Lehrstuhl fu¨r Hochfrequenztechnik Petr LorenzThe TLM method, as considered here, is a numerical time-domain tech- nique which has been used since its introduction by Johns and Beurle

Lehrstuhl fur Hochfrequenztechnikder Technischen Universitat Munchen

Transmission Line Matrix – Multipole Expansion Method

Petr Lorenz

Vollstandiger Abdruck der von der Fakultat fur Elektrotechnik und Infor-mationstechnik der Technischen Universitat Munchen zurErlangung desakademischen Grades eines

Doktor-Ingenieurs

genehmigten Dissertation.

Vorsitzender: Univ.-Prof. Paolo Lugli, Ph.D.

Prufer der Dissertation:1. Univ.-Prof. Dr. techn. Peter Russer2. Prof. Dr. Wolfgang J. R. Hoefer

University of Victoria/Kanada

Die Dissertation wurde am 20.02.2007 bei der Technischen UniversitatMunchen eingereicht und durch die Fakultat fur Elektrotechnik und In-formationstechnik am 02.07.2007 angenommen.

Transmission Line Matrix –Multipole Expansion Method

Petr Lorenz

Institute for High-Frequency Engineeringat the Technische Universitat Munchen

For my wifeand my parents

Acknowledgements

First of all, I would like to thank to my “Doktorvater” Prof. Dr. techn.Peter Russer for giving me the opportunity to work on such interesting andchallenging topic at the Institute for High-Frequency Engineering, and forhis time spent during for me so precious discussions. Thanksto him I couldlearn and enjoy the scientific life, even in the not so easy days of today’sscience.

I want also to thank to my colleagues at the Institute and all the studentswith whom I have spent many hours by discussing the topic.

In my private life I can not thank enough to my wife Ivana for her sup-port, patience, tolerance, those many weekends I have spentworking onthe topic and her love.

Furthermore, I want to thank to my parents and my grandmotherformaking it possible for me to get to know Germany and for encouragingand supporting me in my academic education. Also, many thanks go to allthe others of the family and friends who were always supporting me in mywork.

Lenggries, February 2007

v

Abstract

In this thesis the Transmission Line Matrix — Multipole Expansion method(TLMME) is presented. The TLMME method allows an efficient and po-tentially exact modeling of radiating electromagnetic structures. In theTLMME the time-domain Transmission Line Matrix (TLM) method iscombined with the multipole expansion (ME) method of the radiated field.

The total radiated field is decomposed into orthonormal radiation modeswhich are connected to the TLM simulation domain on a common spheri-cal boundary. In a global network model the simulation domain is modeledby the TLM mesh of transmission lines, every impedance of theradiationmode is modeled by a ladder network one port and the connection of thesepartial networks is accomplished by a connection subnetwork consistingof ideal transformers. This allows to include potentially exact radiatingboundary condition into the TLM model by lumped element equivalentcircuits representing the impedances of radiation modes.

vii

Contents

1 Introduction and Overview 11.1 The Big Picture . . . . . . . . . . . . . . . . . . . . . . . 21.2 Numerical Solutions of Maxwell’s Equations . . . . . . . 41.3 Organization of the Thesis . . . . . . . . . . . . . . . . . 7

2 Transmission Line Matrix (TLM) Method 92.1 Principles of TLM . . . . . . . . . . . . . . . . . . . . . . 92.2 Statistical Mechanics, Diffusion Equation, and TLM . . . 11

2.2.1 One-dimensional diffusion equation . . . . . . . . 112.2.2 Two- and three-dimensional diffusion equation . . 132.2.3 One-dimensional wave equation . . . . . . . . . . 14

2.3 TLM Scattering Matrix in View of Discrete DifferentialForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Two-dimensional TLM cell . . . . . . . . . . . . 192.3.2 Three-dimensional TLM cell . . . . . . . . . . . . 30

2.4 Discrete Source Modeling . . . . . . . . . . . . . . . . . 342.4.1 Example – canonic parallel connection network . . 39

3 Multipole Expansion 413.1 Radiation Modes . . . . . . . . . . . . . . . . . . . . . . 453.2 Cauer Realization of Radiation Modes . . . . . . . . . . . 47

4 Connection Subnetwork 554.1 Tellegen’s Theorem and Connection Subnetwork . . . . . 55

4.1.1 Network form of Tellegen’s theorem . . . . . . . . 554.1.2 Field form of Tellegen’s theorem . . . . . . . . . . 574.1.3 Connection subnetwork . . . . . . . . . . . . . . 57

ix

Contents

4.2 Scattering Matrix of the Connection Subnetwork . . . . . 614.2.1 Direct derivation . . . . . . . . . . . . . . . . . . 614.2.2 Implementation Issues . . . . . . . . . . . . . . . 66

5 Transmission Line Matrix – Multipole Expansion Method 695.1 Spherical TLM Region . . . . . . . . . . . . . . . . . . . 69

5.1.1 Connection of the simulation domain to the radia-tion modes . . . . . . . . . . . . . . . . . . . . . 70

5.2 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . 725.2.1 Cartesian mesh generation . . . . . . . . . . . . . 745.2.2 Wave Digital Filter models of radiation modes . . 76

5.3 The TLMME Algorithm . . . . . . . . . . . . . . . . . . 80

6 Numerical Examples 856.1 Flat Dipole Antenna . . . . . . . . . . . . . . . . . . . . . 85

6.1.1 Analytical characterization . . . . . . . . . . . . . 866.1.2 TLM and TLMME computations . . . . . . . . . 87

6.2 Dipole Antenna at the Boundary of Simulation Region . . 906.3 Bowtie Antenna . . . . . . . . . . . . . . . . . . . . . . . 93

7 Conclusions 99

A Useful Formulas and Relations 103A.1 Central Difference Scheme . . . . . . . . . . . . . . . . . 103A.2 Transformations Between Wave and Network Quantities . 103A.3 Matrix Algebra Rules . . . . . . . . . . . . . . . . . . . . 105

A.3.1 Transposition and inversion . . . . . . . . . . . . 105A.3.2 Block matrix and Schur complement . . . . . . . . 105

A.4 Taylor Expansion in 2D . . . . . . . . . . . . . . . . . . . 106

x

1 Introduction and Overview

We live in time where the demand for electronic devices, electronic instru-ments, electronic systems and electronic information processing devices isbooming. Not only became these devices indispensable toolsin everydaylife but our society became dependent on them. To name just a few exam-ples: it is “impossible” to imagine air traffic control without radio commu-nication, medical surgery without an electrocardiograph (ECG), informa-tion processing without an electronic computer or information spreadingwithout the Internet.

The underlying fabric of all these devices and systems are electric chargesand with them associated electromagnetic fields. It is thus no wonder thatthere is a high demand for tools which let us better understand, optimize,design and synthesize electronic devices and systems. Numerical methodsfor electromagnetics are such tools. With them we obtain an approximativesolution of the electromagnetic equations.

The physical model of electromagnetic phenomena has been completedby James Clerk Maxwell in the year 1865 in his paper “A Dynamical The-ory of the Electromagnetic Field”. The four electromagnetic equations arecalled in honour of Maxwell’s work Maxwell’s equations. In his workMaxwell has introduced the displacement currentJD,

JD =∂D∂t

as a temporal rate of change of the electric flux densityD. The displace-ment current has established a coupling between the electric and the mag-netic field.

However, even if the theory of electromagnetism is 142 yearsold now1,

1In 2007.

1


the solutions of the equations of electromagnetic field can be found ana-lytically for a very limited set of problems. Actually, mostof technicallyinteresting problems do not have an analytic solution, or wedo not knowhow to find it. In these cases we seek for an approximate solution usingnumerical methods.

Numerical methods and approximate solutions of electromagnetic equa-tions have been gaining substantially on importance since about 40 years.The developments of powerful electronic computers in the last years havemade it possible to solve numerically large problems at low cost. Today,full-wave2 solutions of many technically interesting problems can be ob-tained using a personal computer (PC).

Special care must be taken if open radiating electromagnetic structuresare to be characterized with a volumetric discretization method (more onthe classification of numerical methods can be found in Section 1.2). Thisis so, because the computational resources are limited and we can create anumerical model for only a relatively small vicinity of the physical struc-ture. The radiating properties are to be modeled by appropriate boundaryconditions.

Numerical modeling of radiating boundary condition for theTransmissionLine Matrix method is the focus of this thesis.

1.1 The Big Picture

In this thesis the Transmission Line Matrix (TLM) method is the numeri-cal method of choice for one part of the spatial domain. Another part ofthe spatial domain is represented by multipole expansion (ME) technique.The two methods are combined using network oriented approach which isdiscussed in [1]. The matching of the electromagnetic field represented byTLM and ME can be seen as a mode matching technique.

The TLM method, as considered here, is a numerical time-domain tech-nique which has been used since its introduction by Johns andBeurle [2]in 1971 to solve various problems in electromagnetic engineering [3]. In

2Full-wave solution is a solution for both, the electric and magnetic field, in space and time.

2

1.1 The Big Picture

TLM the field is discretized in space and time and modeled by wave pulsespropagating and being scattered in a mesh of transmission lines. Whenradiating electromagnetic structures are modeled the appropriate radiatingboundary conditions at the boundary of the computational domain need tobe included in the computation. In the literature methods torealize absorb-ing boundary conditions (ABCs) are discussed [4], [5], [6],[7], [8], [9].These methods give approximate solutions of the problem by attenuatingthe radiating electromagnetic field, possibly without spurious reflectionsfrom the boundaries of the simulation domain.

On the other hand, the hybrid Transmision Line Matrix – Multipole Ex-pansion (TLMME) method, which is described in this text, yields a po-tentially exact solution for general radiating electromagnetic structures inhomogeneous media. In contrast to the ABC approach, the TLMME mod-els the complete radiated field.

In this hybrid method the problem space is divided in two subspacesR1 andR2 connected at a common boundaryS – see also Figure 1.1.SubspaceR1 represents the TLM region where the discretized structureand the sources are modeled and subspaceR2 represents the homogeneousbackground space where the radiating electromagnetic fieldis representedin terms of ME.

The connection between subspacesR1 andR2 is established from itsnetwork representation. The connection network, which connects subspaceR1 andR2 and contains only ideal transformers [10], [1], is connected tolumped element equivalent circuits representing impedances of radiationmodes.

The basis one-forms [11] of the ME are orthogonal and form a com-plete set. A compact lumped element LC ladder network representing thecoupling between the electric and magnetic field intensities may be estab-lished [10].

In the TLM method the spherical boundary is approximated by cubi-cal TLM cells. The connection of the outer faces of the TLM cells onthe spherical boundary to the spherical modes is modeled by aconnectionsubnetwork.

3


TE11

TM11

TEm’n’

TMm"n"

TLM TLM

R2R1R1

S

Figure 1.1: Left: schematic representation of the divisionof the unboundedspace in subspacesR1 andR2 with the boundaryS . Right:schematic representation of the global network model of theTLMME simulation.

1.2 Numerical Solutions of Maxwell’sEquations

In this Section a brief classification of main numerical methods used forsolving the electromagnetic equations is given and the reason for the trun-cation of the simulation domain is shown.

For the numerical solution of Maxwell’s equations various techniqueshave been proposed and new are still coming. We can distinguish be-tween time-domain and frequency-domain techniques. Another possibil-ity is to distinguish between volumetric discretization methods and surfacediscretization methods (see Figure 1.2). Volumetric discretization meth-ods lead to finite-difference formulations, whereas surface discretizationmethods to integral equation formulations.

As we can see, there is a vast number of numerical methods one canuse to solve the electromagnetic problem. However, not every method issuitable for a particular problem. A hybrid approach is an attractive wayhow to combine the power of different methods.

4

1.2N

umericalS

olutionsofM

axwell’s

Equations

Numerical methods in electromagnetics

Time domain Frequency domain

Surface discretization Volumetric discretization

IntegralEquation

Methodof

Moments

TransmissionLine

Matrix

FiniteDifference

FiniteIntegral

FiniteElement

FiniteVolume

Fig

ure

1.2

:Classificatio

no

fmain

nu

mericalm

etho

ds

inelectro

mag

netics.

5


Truncated simulation domains

Since the computational resources of today’s computers arevery limited, itis impossible to discretize large volumes of space. The limits up to whichwe are able to solve a problem are set by

• memory requirements,

• time needed to process the information.

To get a better feeling for what are the memory requirements to solvean electromagnetic problem, let us take a look at the following example.There is given a structure assumed to be perfect electric conductor (PEC)in air and we wish to compute the scattered electromagnetic field. A three-dimensional computational domain of 2000× 2000× 2000 TLM cells re-quires 3072000000000 bits3 to hold the information about the electro-magnetic field. This amount of information means

2000×2000×2000×12×321024×1024×1024×8

= 357.63GB of memory.

Increasing the spatial resolution just twice in every direction results in eighttimes larger memory requirements. For the particular example, 2861GB ofmemory would be needed.

Consider that the human genome needs 70000000 bits to hold its infor-mation. The previous example needs more than 40000 times more. Fur-thermore, the time needed to process this amount of information is inverseproportional to the speed of information processing.

This example shows the reasons why we are not able to discretize verylarge structures. Consequently, the computational regionmust be truncatedto finite, relatively small region around the physical structure. At the outerboundaries of this computational region special boundary conditions needto be applied.

3For the modeling of free-space we need 12 variables per TLM cell. We assume a TLMvariable to be represented by a 32 bit floating point number.

6

1.3 Organization of the Thesis

For radiating structures the special boundary condition iscalledradiat-ing boundary condition; for periodic structuresperiodic boundary condi-tion is used; for resonating structures assumed to be inside a perfect elec-tric conducting boxPEC boundaryconditions are applied; the dual perfectconducting magnetic wallsPMC boundary conditionsare used usually toimpose symmetry conditions.

Of the mentioned boundary conditions the radiating boundary condi-tions are the most difficult ones to be realized numerically.

1.3 Organization of the Thesis

The thesis is organized as follows. In Chapter 2 the Transmission LineMatrix (TLM) method is described in detail. The principles of the methodare shown from the point of view of statistical mechanics andthe relationof the TLM scattering matrix to Maxwell’s equations is expressed in termsof discrete differential forms. A way of modeling discrete sources suitablefor the modeling of excitation of antenna structures is presented.

Chapter 3 is devoted to the multipole expansion (ME) method.The radi-ation modes and their canonical circuit realizations are described in detail.

Chapter 4 discusses the connection between the TLM domain and theouter domain represented in terms of ME. The fundamental Tellegen’s the-orem is described and the scattering matrix of the connection subnetworkis derived.

The hybridization of the TLM method and the ME method is presentedin Chapter 5. First, the spherical TLM domain needed for the TLMMEmethod is described in detail. Next, the pre-processing related tasks arediscussed. The generation of the connection matrix and the realization ofthe canonical equivalent circuits of the radiation modes using wave digitalfilters (WDFs) are shown. The Chapter is concluded with the descriptionof the TLMME algorithm.

Chapter 6 presents examples of electromagnetic solutions to antennastructures using the TLMME method and shows the validity, accuracy andnew features of the method.

7


Finally, the conclusions of the thesis are presented in Chapter 7. Usefulformulas, relations and operations are listed in Appendix A.

8

2 Transmission Line Matrix(TLM) Method

2.1 Principles of TLM

The Transmission Line Matrix (TLM) method is a discrete model, in spaceand time, of the electromagnetic phenomena.

The TLM method has been introduced by Johns in 1974 as a two di-mensional method [12] and extended in 1987 for three dimensions withthe development of symmetrical condensed node [13]. Since then therehave been many attempts to derive the method directly from Maxwell’sequations, and so to prove the validity of the method [14], [15], [16], [17],[18], [19].

The way from the continuous model to the discrete one is the standardway used to derive finite difference time domain (FDTD) method, finiteintegral technique (FIT) and other methods. Also, discretedifferential cal-culus follows this way.

On the other hand, as Toffoli suggests (see [20] and [21]), the other wayaround, i.e., from a discrete model to the continuous one, ismuch more ap-propriate, if we want to develop intrinsically discrete models of nature. Theargument, why we should do this, is supported by the rapid developmentof digital technologies and consequent information processing facilities.These make it possible to have new ways of describing the nature.

It is difficult, however, to cope with the idea of giving up the powerfultechniques and models developed during the last 300 years represented bymeans of differential equations. The models have, however, their limitsandmost of technically interesting phenomena can not be solvedby analyticaltechniques directly. But we do not want to give them up. We want to de-

9

2 Transmission Line Matrix (TLM) Method

velop alternative intrinsically discrete models of natural phenomena whichwill behave, under specific circumstances, like the analytical models.

The idea of a transition from a discrete description to a continuous oneis well-known in statistical mechanics. In Section 2.2 we will see how theone-dimensional, two-dimensional and three-dimensionaldiffusion equa-tions and the one-dimensional wave equation are obtained ascontinuumlimits of intrinsically discrete models.

The TLM method can be seen in a broader view as a cellular automa-ton (CA). The notion of cellular automata has been first explored by Johnvon Neumann. Cellular automata consist of independent computing units,which exchange their values with neighboring units and, accordingly, changetheir state. A cellular automaton has intrinsically parallel nature1.

Usually, CA are considered to contain only discrete variables as theirstates. The Coupled Map Lattices (CML) [22] are computational modelswhere the internal state of each cell is represented in termsof continuousvariables, but space and time remain discrete. The computational mecha-nism of CML are identical with those of CA. Thus, the difference betweenCA and CML is in the representation of the state of a cell by a discrete orcontinuous variable, respectively2.

The state of a TLM cell can be described by a vector of wave quantities.The transformation from one state to another – the computation of one cell– is represented by a scattering process. So, if the information about thestate before scattering is stored in a vectora and the result of the scattering– computation – in a vectorb, we may express the computation process ofa TLM cell as

b = Sa, (2.1)

whereS is the scattering matrix. The scattering matrix defines the compu-tation performed by a cell.

In a consequent step the information between neighboring cells is ex-changed and the process starts from the beginning (more detailed descrip-

1With the introduction of multicore processors parallelismis gaining on importance morethan ever.

2CML models have been used recently to implement physically-based visual simulationsfor real-time visual simulations [23].

10

2.2 Statistical Mechanics, Diffusion Equation, and TLM

tion of TLM iteration is given later in Section 5.3). This algorithm is equalto the algorithm of a cellular automaton using continuous state variables.In Section 2.3 it is shown how the TLM scattering matrices of two- andthree-dimensional free-space TLM nodes can be obtained using discretedifferential forms.

2.2 Statistical Mechanics, Diffusion Equation,and TLM

2.2.1 One-dimensional diffusion equation

In this Section it is shown how the behaviour of a discrete system in itscontinuum limit is investigated. In the text I follow the description and Iuse the same notation as in [24] (pp.15–20).

The one-dimensional diffusion equation is obtained as a continuum limitof one-dimensional uncorrelated random walk. A random walkis a processin which a particle moves a distance|±∆x| in time∆t. The direction of themovement, i.e., in the positive or negativex-direction, is random. Bothcases occur with the same probability.

We can describe the random walk by a probability distribution functionχ(x) defined for the particular case of one-dimension as

χ(x) =12δ(x−∆x)+

12δ(x+∆x), (2.2)

whereδ(x) is the Dirac delta function. The first three moments ofχ(x) are∫

χ(x) dx= 1, (2.3a)∫

xχ(x) dx= 0, (2.3b)∫

x2χ(x) dx= (∆x)2. (2.3c)

We see from (2.3) that the mean value ofχ(x) is zero and that the standarddeviation is∆x.

11


The one-dimensional diffusion equation is given by the equation

∂ρ(x, t)∂t

= D∂2ρ(x, t)

∂x2, (2.4)

whereD is the diffusion coefficient. The diffusion equation describes theevolution of the density functionρ(x, t).

The density functionρ(x, t) can be interpreted differently for a one par-ticle system and a many particle system. For a one particle system thedensity function describes the evolution of the probability density for lo-cating the particle at positionx at time t. For many particle system thedensity function describes the density of diffunding particles (mass).

If we know the probability densityρ(x, t) at a particular timet, its evolu-tion at timet+∆t can be written with (2.2) as

ρ(x, t+∆t) =∫

χ(x− x′)ρ(x′, t) dx′ =∫

χ(z)ρ(x−z, t) dz, (2.5)

with z= x− x′. Taking the Taylor expansion ofρ(x−z, t) aroundx we canwrite (2.5) as

ρ(x, t+∆t) =∫

χ(z)

[

ρ(x, t)−z∂ρ(x, t)∂x

+z2

2∂2ρ(x, t)

∂x2+ . . .

]

dz. (2.6)

Using only the first three terms of the Taylor expansion in (2.6) we obtainwith (2.3) the approximative expression

ρ(x, t+∆t) ≈ ρ(x, t)+(∆x)2

2∂2ρ(x, t)

∂x2. (2.7)

Reorganizing (2.7) and dividing both sides by∆t we obtain the approxima-tive equation

1∆t

(

ρ(x, t+∆t)−ρ(x, t))

≈ (∆x)2

2∆t∂2ρ(x, t)

∂x2. (2.8)

In the continuum limit this equation becomes the diffusion equation (2.4)with the diffusion coefficientD given by

D =(∆x)2

2∆t. (2.9)

12


The random walk can be implemented numerically by an algorithm sim-ilar to the TLM algorithm (the CA algorithm). From (2.2) we know thatif a particle arrives at timet at a nodexm, there is equal probability that attime t+∆t this particle will be either at the nodexm−1 or at the nodexm+1.The scattering matrix

S=12

[

1 11 1

]

(2.10)

models then the one-dimensional diffusion equation.

2.2.2 Two- and three-dimensional diffusion equation

In a similar way to the previous Section we may derive the two-dimensionaldiffusion equation given by

∂ρ(x,y, t)∂t

= D

(

∂2ρ(x,y, t)

∂x2+∂2ρ(x,y, t)

∂y2

)

. (2.11)

The probability distribution for the two-dimensional random walk takesthe form

χ(x,y) =14δ(x−∆x,y)+

14δ(x+∆x,y)+

14δ(x,y−∆y)+

14δ(x,y+∆y).

(2.12)Using the two-dimensional Taylor expansion (see Appendix A, Equation(A.18)) it is possible to show that the two-dimensional random walk in itscontinuum limit corresponds to the two-dimensional diffusion equation.

Similarly to the one-dimensional case, the scattering matrix

S=14

1 1 1 11 1 1 11 1 1 11 1 1 1

(2.13)

13


models the two-dimensional diffusion equation and the scattering matrix

S=16

1 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 1

(2.14)

models the three-dimensional diffusion equation.

2.2.3 One-dimensional wave equation

The scattering matrix of the one-dimensional TLM method in free-space isgiven by

S=[

0 11 0

]

. (2.15)

We will see now that with this scattering matrix the one-dimensional waveequation is modeled. It is easy to check that the scattering matrix is energyconservative (|det(S)| = 1) and invariant to time-reversal (S·S= I ).

The one-dimensional wave equation is given by

∂2ρ(x, t)

∂t2= c2∂

2ρ(x, t)

∂x2, (2.16)

where the constantc is the speed of propagation.

Using the scattering matrix (2.15) the probability distribution functionχ(x) can be written as

χ(x) = δ(x±∆x). (2.17)

14


The first moments of the probability distribution are∫

χ(x) dx= 1, (2.18a)∫

xχ(x) dx= ∓∆x, (2.18b)∫

x2χ(x) dx= (∆x)2, (2.18c)∫

x3χ(x) dx= ∓(∆x)3, (2.18d)∫

x4χ(x) dx= (∆x)4, (2.18e)

... (2.18f)

In analogy to (2.6) and by using (2.17) we get

ρ(x, t+∆t) =∫

χ(z)ρ(x−z, t) dz=∫

ρ(x−z, t)δ(z±∆x) dz= ρ(x±∆x, t)

(2.19)which is the one-dimensional advection equation for the density functionρ(x, t). In (2.19) we have used the sampling property of the Dirac deltafunction. From these two solutions we obtain directly the one-dimensionalwave equation.

If we interpret the probability distributionρ(x, t) as wave quantities, wesee from (2.19) that the wave quantities satisfy exactly

a(x, t+∆t) = a(x−∆x, t), (2.20a)

b(x, t+∆t) = b(x+∆x, t). (2.20b)

Using Taylor expansion we may approximate (2.20) by

a(x−∆x, t) = a(x, t)−∆x∂a(x, t)∂x

+O(∆x2), (2.21a)

b(x+∆x, t) = b(x, t)+∆x∂b(x, t)∂x

+O(∆x2), (2.21b)

15


which may be rewritten as (using approximations for smoothly varyingquantities in space and time)

a(x, t+∆t)−a(x, t)∆t

≈ ∂a(x, t)∂t

≈ −∆x∆t

∂a(x, t)∂x

, (2.22a)

b(x, t+∆t)−b(x, t)∆t

≈ ∂b(x, t)∂t

≈ ∆x∆t

∂b(x, t)∂x

. (2.22b)

These are the one-dimensional advection equations for the wave quantitiesa(x, t) andb(x, t).

Using the transformations (A.5b) from wave quantities to network quan-tities (see Appendix A) withη = 1 we have the following relations

V(x, t) = a(x, t)+b(x, t), (2.23a)

I (x, t) = a(x, t)−b(x, t). (2.23b)

Using (2.22) and (2.23) and definingc = ∆x∆t we obtain the following

equations forV(x, t) andI (x, t)

∂V(x, t)∂x

=∂a(x, t)∂x

+∂b(x, t)∂x

= −1c

(

∂a(x, t)∂t

− ∂b(x, t)∂t

)

= −1c∂I (x, t)∂t

,

(2.24a)

∂I (x, t)∂x

=∂a(x, t)∂x

− ∂b(x, t)∂x

= −1c

(

∂a(x, t)∂t

+∂b(x, t)∂t

)

= −1c∂V(x, t)∂t

,

(2.24b)

which are the well-known Telegrapher’s equations for voltage and currenton a transmission line.

2.3 TLM Scattering Matrix in View of DiscreteDifferential Forms

The scattering matrix of the 2D TLM shunt node is equivalent to the scat-tering matrix of the parallel adaptor used in wave digital filters (WDFs)

16

2.3 TLM Scattering Matrix in View of Discrete Differential Forms

[25]. In this section this equivalence is examined and it will be shown thatalso the 2D TLM scattering matrix with stubs can be obtained from the par-allel adaptor of WDFs. The result can be generalized to obtain valid (byvalid allowable is meant) TLM scattering matrices satisfying the followingproperties:

1. Frequency independent,

2. Invariant to time-reversal (S·S= I ),

3. Energy conservative (|det(S)| = 1),

4. Eigenvalues equal±1 (i.e., it is a connection matrix).

A standard 2D TLM shunt node is shown in Figure 2.1. The character-istic impedance of the link lines isZ0 =

√L/C. The interconnection at the

center of the node will be calledcell center. The cell center is delay-free,frequency independent and energy conservative. The scattering matrix ofa shunt cell center equals to the scattering matrix of a parallel adaptor ofWDF.

A 2D TLM cell center with a shunt stub line connection is showninFigure 2.2. The reference impedances of the TLM link lines are Z0 =

1Y0

and the reference impedance of the stub line isZs =1Ys

. The scatteringmatrix of parallel adaptor for the given cell center can be written as ([25],pp. 276–277)

S=

(γ−1) γ γ γ γs

γ (γ−1) γ γ γs

γ γ (γ−1) γ γs

γ γ γ (γ−1) γs

γ γ γ γ (γs−1)

, (2.25a)

where

4γ+γs= 2 ⇒ γs= 2−4γ (2.25b)

17


Z0 Z0

Z0

Z0

TLM cell center

Figure 2.1: 2D TLM shunt node and its TLM cell center.

Z0 Z0

Z0

Z0

Zs

Figure 2.2: The 2D TLM cell center with a shunt stub line connection withreference impedanceZs.

18


and

γ =2

4+ YsY0

, (2.25c)

γs=2Ys

Y0

4+ YsY0

. (2.25d)

Inserting (2.25c) and (2.25d) into (2.25a) we obtain the expression for thescattering matrix

S=1

Y

(2− Y) 2 2 2 2YsY0

2 (2− Y) 2 2 2YsY0

2 2 (2− Y) 2 2YsY0

2 2 2 (2− Y) 2YsY0

2 2 2 2 (2YsY0− Y)

, (2.25e)

where

Y= 4+Ys

Y0. (2.25f)

The expression (2.25e) is the well-known scattering matrixof the 2D TLMshunt node with a shunt stub line (cf., e.g., [26], p. 97).

2.3.1 Two-dimensional TLM cell

For the relation between electrical network quantities andelectromagneticfield quantities discrete exterior calculus is applied. Discrete exterior cal-culus also introduces the discrete exterior derivative, the discrete counter-part of the exterior derivative operator.

The network quantities are obtained from the electromagnetic quantitiestaking the integrals of them. This is equivalent to what is also called finiteintegrals. However, in contrast to finite integrals, the underlying structureof the manifolds is maintained.

Let us take a look at 2D example. We assume invariance of the electro-magnetic field with respect toz direction, i.e., ∂

∂z = 0. Consequently, only

19


2D problem description is appropriate. Furthermore, the TMz modes areconsidered for now.

Using differential forms notations, the electromagnetic phenomena ofTMz modes in free-space (ε0,µ0) is described with the following reducedMaxwell’s equations

(

∂Hy

∂x− ∂Hx

∂y

)

dx∧ dy= ε0∂Ez

∂tdx∧ dy, (2.26a)

∂Ez

∂xdz∧ dx= µ0

∂Hy

∂tdz∧ dx, (2.26b)

∂Ez

∂ydy∧ dz= −µ0

∂Hx

∂tdy∧ dz. (2.26c)

The unit one-forms dx, dy, dz, their wedge product and the Hodge operatorcan be visualized as shown in Figure 2.3.

dx

dy

dz

dx∧ dy

dx

dy

x

yz

Figure 2.3: Unit one-forms dx, dy, dz and the two-form dx∧ dy= ⋆dz inan Euclidean three-dimensional space.

The discrete counterparts of the unit one-forms are the discrete one-forms∆x, ∆y and∆z. Similarly to the differential forms, we may define forthe discrete forms the wedge product, the Hodge operator andthe discreteexterior derivative operation [27].

We introduce a mapping from electromagnetic field quantities to net-work quantities. The mapping is shown in Figure 2.5. We assign to each

20


∆x

∆y

∆z

∆x∧∆y

∆x

∆y

x

yz

Figure 2.4: Discrete one-forms∆x,∆y,∆zand the two-form∆x∧∆y=⋆∆zin an Euclidean three-dimensional space.

V = −∫

∆xE

∆x

I =∫

∆xH

∆x

x

Figure 2.5: Mapping from electromagnetic field quantities to networkquantities.

21


oriented line of the TLM cell two network quantities:voltageandcurrent.The resulting 2D TLM cell with assigned network quantities is show inFigure 2.6.

I1y

I4x

I3y

I2x

V2z

I1z

V4z

I2z

I4z

V3z

I3z

V1z

x

yz

Figure 2.6: 2D TLM cell with assigned network quantities forTM modes.

The expressions for discrete exterior derivatives of TM field with theintroduced network quantities can be written as

dH|xy = I1y + I4

x − I3y − I2

x, (2.27a)

dH|zx= I1z − I2

z = 0, since∂Iz

∂z= 0, (2.27b)

dH|yz= I4z − I3

z = 0, since∂Iz

∂z= 0, (2.27c)

dE|xy = 0, (2.27d)

dE|zx= V2z −V4

z , (2.27e)

dE|yz= V3z −V1

z . (2.27f)

22


The invariance of macroscopic quantities (field intensities, voltages andcurrents) to the scattering process – with the assumption ofthe scatter-ing process taking place inside the TLM cell center – and the frequency-independence of the scattering matrix imply that∂

∂t = 0 in the Maxwell’sequations (2.26). With this assumption the equations whichmust be satis-fied during the scattering process are

(

∂Hy

∂x− ∂Hx

∂y

)

dx∧ dy= dH|xy = 0, (2.28a)

∂Ez

∂xdz∧ dx= dE|zx= 0, (2.28b)

∂Ez

∂ydy∧ dz= dE|yz= 0. (2.28c)

These are the equations of static electric and magnetic fields.Inserting the expressions for discrete exterior derivatives in terms of net-

work quantities (2.27) into (2.28) we get

I1y + I4

x − I3y − I2

x = 0, (2.29a)

V2z −V4

z = 0 ⇒ V2z = V4

z , (2.29b)

V3z −V1

z = 0 ⇒ V3z = V1

z , (2.29c)

Equation (2.29a) can be identified as Kirchhoff’s current law (KCL) (seealso Figure 2.7). The resulting 2D TLM cell center with assigned networkquantities is show in Figure 2.8.

Since the equations (2.28) describe static fields, any integral on a closedpath must equal to zero. Taking the integral on a closed path as shown inFigure 2.9 results in the following additional condition

V1z −V4

z +V3z −V2

z = 0 ⇒ V1z +V3

z = V4z +V2

z . (2.30)

From (2.29) we know thatV1z = V3

z andV4z = V2

z so we obtain the condition

V1z = V2

z = V3z = V4

z . (2.31)

23


I1y

I2x

I3y

I4x

V1z

V2z

V3z

V4z

G

x

y

Figure 2.7: KCL and KVL for the 2D TLM cell center.

I1y

I4x

I3y

I2x

V2z

0

V4z

0

0

V3z

0

V1z

x

yz

Figure 2.8: 2D TLM cell center with assigned network quantities for TMmodes.

24


P0

V2z

0

V4z

0

0

V3z

0

V1z

x

yz

Figure 2.9: Integration path.

The network quantities may be transformed to wave quantities using thefollowing transformation rules

V1z = a1+b1, (2.32a)

V2z = a2+b2, (2.32b)

V3z = a3+b3, (2.32c)

V4z = a4+b4, (2.32d)

I1y = a1−b1, (2.32e)

I2x = −a2+b2, (2.32f)

I3y = −a3+b3, (2.32g)

I4x = a4−b4. (2.32h)

Inserting (2.32) into (2.29) and (2.31) we obtain the following set of

25


linearly independent equations

a1+a4+a3+a2 = b1+b4+b3+b2, (2.33a)

a3+b3 = a1+b1, (2.33b)

a2+b2 = a4+b4, (2.33c)

a1+b1 = a2+b2. (2.33d)

Instead of (2.33d) we can also take the equation

a3+b3 = a4+b4 (2.33e)

without a change in the result. Defining the vectors of incoming and out-going waves

a=(

a1,a2,a3,a4)T, (2.34a)

b =(

b1,b2,b3,b4)T, (2.34b)

we can write (2.33) asAa = Bb, (2.35)

with

A =

1 1 1 1−1 0 1 00 1 0 −11 −1 0 0

, (2.36a)

B =

1 1 1 11 0 −1 00 −1 0 1−1 1 0 0

. (2.36b)

Using the definition of scattering matrix, i.e.,b = Sa, and with (2.35),we see that the scattering matrix can be computed from matricesA andBas follows

S= B−1A. (2.37)

26


With the inverse ofB given by

B−1 =14

1 1 −1 −21 1 −1 21 −3 −1 −21 1 3 2

(2.38)

we obtain the scattering matrix

S=14

1 1 −1 −21 1 −1 21 −3 −1 −21 1 3 2

·

1 1 1 1−1 0 1 00 1 0 −11 −1 0 0

=12

−1 1 1 11 −1 1 11 1 −1 11 1 1 −1

,

(2.39)

which is the well-known scattering matrix of the 2D shunt TLMnode (cf.,e.g., [26], p. 95).

Dual to the TM modes are TE modes. The derivation of the scatteringmatrix for 2D TE modes follows the same steps as for the TM modes andwill be presented briefly. The 2D TLM cell center for TE modes is shownin Figure 2.10.

The independent set of linear equations obtained from the equationsdE = 0 and dH = 0 using the discrete exterior calculus on regular rect-angular mesh is given by

V1x +V4

y −V3x −V2

y = 0, (2.40a)

I3y − I1

y = 0, (2.40b)

I2x − I4

x = 0, (2.40c)

I1y − I4

x+ I3y − I2

x = 0. (2.40d)

Equation (2.40a) is derived from dE = 0 and equations (2.40b) and (2.40c)are derived from dH = 0. The derivation of (2.40d) follows the sameprocedure of deriving (2.30).

27


V1x

V4y

V3x

V2y

I2x

0

I4x

0

0

I3y

0

I1y

x

yz

Figure 2.10: 2D TLM cell center with assigned network quantities for TEmodes.

28


The transformation from network quantities to wave quantities is definedby

V1x = a1+b1, (2.41a)

V2y = a2+b2, (2.41b)

V3x = a3+b3, (2.41c)

V4y = a4+b4, (2.41d)

I1y = −a1+b1, (2.41e)

I2x = a2−b2, (2.41f)

I3y = a3−b3, (2.41g)

I4x = −a4+b4. (2.41h)

With the definitions (2.34) and (2.41), equations (2.40) arewritten in theform of (2.35), where

A =

1 −1 −1 11 0 1 00 1 0 1−1 −1 1 1

, (2.42a)

B =

−1 1 1 −11 0 1 00 1 0 1−1 −1 1 1

(2.42b)

and the inverse matrix of the matrixB is

B−1 =14

−1 2 0 −11 0 2 −11 2 0 1−1 0 2 1

. (2.42c)

29


The scattering matrix obtained fromS= B−1A reads

S=12

1 1 1 −11 1 −1 11 −1 1 1−1 1 1 1

, (2.43)

which is the scattering matrix of the 2D series TLM node (cf.,e.g., [26],p. 78).

2.3.2 Three-dimensional TLM cell

The scattering matrix of the 3D symmetrical condensed TLM node is ob-tained using the same techniques as in the previous section.However, inthe 3D case no decomposition into TE and TM field is possible. Conse-quently, the field intensities are considered with all components, i.e.,

E = Ex dx+Eydy+Ezdz, (2.44a)

H = Hx dx+Hydy+Hzdz. (2.44b)

The network quantities are obtained from the electromagnetic field quan-tities in the same way as described in the two-dimensional case. The 3DTLM cell is shown in Figure 2.11.

The following are the 12 linearly independent equations which must be

30


V7,I5

V2,I4

V8,I6

V1,I3

V3,I1

V10,I12

V4,I2

V9,I11

V11,I9

V6,I8

V12,I10

V5,I7

x

yz

Figure 2.11: 3D TLM cell with assigned network quantities.

31


satisfied by the network quantities inside the symmetrical condensed node.

V7+V2−V8−V1 = 0, (2.45a)

I5+ I4− I6− I3 = 0, (2.45b)

V11+V6−V12−V5 = 0, (2.45c)

I9+ I8− I10− I7 = 0, (2.45d)

V3+V10−V4−V9 = 0, (2.45e)

I1+ I12− I2− I11= 0, (2.45f)

V5+V6−V4−V3 = 0, (2.45g)

V7+V8−V10−V9 = 0, (2.45h)

V1+V2−V12−V11= 0, (2.45i)

I7+ I8− I1− I2 = 0, (2.45j)

I5+ I6− I12− I11= 0, (2.45k)

I3+ I4− I10− I9 = 0. (2.45l)

The transformations from wave quantities to network quantities are de-fined by the following equations.

Vi = ai +bi , (2.46a)

32


wherei = 1. . .12 and

I1 = a1−b1, (2.46b)

I2 = −a2+b2, (2.46c)

I3 = −a3+b3, (2.46d)

I4 = a4−b4, (2.46e)

I5 = a5−b5, (2.46f)

I6 = −a6+b6, (2.46g)

I7 = −a7+b7, (2.46h)

I8 = a8−b8, (2.46i)

I9 = a9−b9, (2.46j)

I10= −a10+b10, (2.46k)

I11= −a11+b11, (2.46l)

I12= a12−b12. (2.46m)

The matricesA andB for the three-dimensional case read

A =

−1 1 0 0 0 0 1 −1 0 0 0 00 0 1 1 1 1 0 0 0 0 0 00 0 0 0 −1 1 0 0 0 0 1 −10 0 0 0 0 0 1 1 1 1 0 00 0 1 −1 0 0 0 0 −1 1 0 01 1 0 0 0 0 0 0 0 0 1 10 0 −1 −1 1 1 0 0 0 0 0 00 0 0 0 0 1 1 −1 −1 0 0 01 1 0 0 0 0 0 0 0 0 −1 −1−1 1 0 0 0 0 −1 1 0 0 0 00 0 0 0 1 −1 0 0 0 0 1 −10 0 −1 1 0 0 0 0 −1 1 0 0

,

(2.47a)

33


B=

1 −1 0 0 0 0 −1 1 0 0 0 00 0 1 1 1 1 0 0 0 0 0 00 0 0 0 1 −1 0 0 0 0 −1 10 0 0 0 0 0 1 1 1 1 0 00 0 −1 1 0 0 0 0 1 −1 0 01 1 0 0 0 0 0 0 0 0 1 10 0 1 1 −1 −1 0 0 0 0 0 00 0 0 0 0 −1 −1 1 1 0 0 0−1 −1 0 0 0 0 0 0 0 0 1 1−1 1 0 0 0 0 −1 1 0 0 0 00 0 0 0 1 −1 0 0 0 0 1 −10 0 −1 1 0 0 0 0 −1 1 0 0

(2.47b)

The scattering matrix of the 3D TLM node for free-space is computedfrom S= B−1A and reads

S=12

0 0 0 0 0 0 1 −1 0 0 1 10 0 0 0 0 0 −1 1 0 0 1 10 0 0 0 1 1 0 0 1 −1 0 00 0 0 0 1 1 0 0 −1 1 0 00 0 1 1 0 0 0 0 0 0 1 −10 0 1 1 0 0 0 0 0 0 −1 11 −1 0 0 0 0 0 0 1 1 0 0−1 1 0 0 0 0 0 0 1 1 0 00 0 1 −1 0 0 1 1 0 0 0 00 0 −1 1 0 0 1 1 0 0 0 01 1 0 0 1 −1 0 0 0 0 0 01 1 0 0 −1 1 0 0 0 0 0 0

(2.48)

2.4 Discrete Source Modeling

For the modeling of antenna problems discrete sources are ofimportance.Since the TLMME method is applied to radiating problems, a suitable

34


modeling of discrete sources is needed. In this section the source portbased on wave digital filters andboundary oriented field mapping[16] isdescribed and it will be shown how it can be applied to model a discretesource distributed over more TLM cells.

As was shown in Section 2.3, the 2D TLM method and in particularthe scattering matrix are related to the parallel and seriesadaptors of wavedigital filters. Here we use this relation to describe a discrete port for theTLM method.

The connection network based on parallel and series adaptors of WDFscan be used for the modeling of discrete sources. The connection networkis anN+ 1 port network withN TLM portsconnected to the TLM meshand onesource port. The source port is the input port to the TLM model.

The TLM ports of the connection network are connected to the TLMlink lines at the interfaces between adjacent TLM cells. Consequently,the mapping type between network and field quantities is the boundaryoriented field mapping. Schematic pictures of canonic parallel and seriesconnection networks exciting one field polarization are shown in Figure2.12 and Figure 2.13.

TLM TLMZl1 Zl2

Zp Link line

Parallel adaptor

Source port

Figure 2.12: Canonic parallel connection subnetwork; the referenceimpedances of the TLM link lines areZl1 andZl2 and the ref-erence impedance of the source port isZp.

35


TLM TLMZl1 Zl2

ZpLink line

Series adaptor

Source port

Figure 2.13: Canonic series connection subnetwork; the referenceimpedances of the TLM link lines areZl1 andZl2 and the ref-erence impedance of the source port isZp.

We can see that the discrete voltage port for one polarization can be seenas parallel adaptor placed at the interface of two TLM link lines.

The parallel connection network has the following properties:

1. A common voltage (i.e., electric field intensity) is impressed at theinterface between adjacent TLM cells.

2. Different line impedances of the TLM link lines and the source portare taken into account.

3. The scattering matrix of the connection network is

• real,

• frequency independent,

• energy conservative,

• time-reversible and

• the eigenvalues equal±1. If all the TLM port impedances andthe source port impedance are equal then the scattering matrixis also unitary and symmetric.

36


A series connection network is dual to the parallel connection network.The properties of the series connection network are the sameas those of theparallel one except that a common current (i.e., magnetic field intensity) isimpressed at the interface between adjacent TLM cells.

A series port is dual to the parallel port and is shown in Figure 2.13. Theproperties of the series discrete port are

1. A common current (i.e., magnetic field intensity) is impressed at theinterface.

2. Different line impedances of the link lines and the discrete portaretaken into account by the adaptor.

3. The series adaptor is characterized by a (3×3) scattering matrix withthe same properties as for the parallel adaptor

To the discrete port three different source types can be connected. Theseare

1. ideal voltage source,

2. ideal current source,

3. power source with defined internal impedance.

The different source types are shown in Figure 2.14.

(a) (b) (c)

Zs

GsV0 I0 V0 I0

Figure 2.14: Lumped sources which may be connected to the discrete portadaptor.

37


The WDF realizations of these sources can be connected directly to thesource port of the connection network. The WDF realizationsof the dif-ferent source types with wave digital filters are shown in Figure 2.15. Weuse here the same convention for symbols of WDF adaptors and wave flowdiagrams as in [25].

replacements

2V0

−1

a

b2RsI0

a

bV0; RsI0

a

b

(a) (b) (c)

Figure 2.15: WDF representations of the sources.

The power source can also be seen (using Thevenin theorem) as an idealvoltage source connected at infinity to one end of a transmission line withimpedanceZS. The other end of the transmission line is connected to thediscrete port.

The discrete port just described implements following features into theTLM algorithm:

• Well-defined excitation of field amplitudes and energy.

• Impulsive characterization of electromagnetic structures with linearproperties. This is possible since only one port is available from“outside” and the TLM model is linear.

• Direct calculation of scattering parameters from the incoming andreflected wave amplitudes.

An example on impulsive excitation of bowtie antenna is given in Sec-tion 6.3.

38


2.4.1 Example – canonic parallel connection network

The scattering matrix referred to the port impedanceZp =3772 Ω with Zl1 =

Zl2 = 377Ω reads

Sp =

0 12

12

1 −12

12

1 12 −1

2

. (2.49)

The scattering matrix referred to the port impedanceZp = 50Ω with Zl1 =

Zl2 = 377Ω reads

Sp =

0.58071 0.20964 0.209641.58071 −0.79036 0.209641.58071 0.20964 −0.79036

. (2.50)

It is easy to check thatSpSp = I and|det(Sp)| = 1. The matrices are thusvalid scattering matrix of the TLM method.

39

3 Multipole Expansion

In the multipole expansion the total radiated electromagnetic field is ex-pressed as linear combinations of known basis one-forms, called radiationmodes. In homogeneous medium the radiation modes form a completeorthonormal set of basis one-forms. These are considered tohave an in-finitesimally small source in the center of the coordinate system used forthe expansion which makes them behave singularly there. Hence the namemultipole expansion.

For the expansion we use the spherical coordinate system (r,ϑ,ϕ) – (ra-dius, elevation, azimuth) – see also Figure 3.1. We decompose the to-tal field into transverse and longitudinal partsN = N t +Nr dr with N t =

Nϑr dϑ+Nϕr sinϑdϕ, whereN stands for the one-form field intensity pha-sorsE orH . Phasor quantities are distinguished by underlined font.

It is shown in [28], that it is sufficient to expand only the transverse partsof the total field. These are the independent field quantitiesfrom which theradial (longitudinal) components can be computed.

Based on these considerations we write the tangential field intensities as

Et(r,ϑ,ϕ, t) =∑

p

Vp(r, t)ep(ϑ,ϕ), (3.1a)

Ht(r,ϑ,ϕ, t) =∑

p

Ip(r, t)hp(ϑ,ϕ) =∑

p

Ip(r, t)(⊥n ep(ϑ,ϕ)

)

, (3.1b)

whereVp(r, t) and Ip(r, t) are generalized voltagesand generalized cur-rents, respectively, andep(ϑ,ϕ) andhp(ϑ,ϕ) are the transverse basis one-forms of the electric field intensity and the magnetic field intensity, respec-tively, corresponding to the modep. The mode indexp summarizes fourindices (n,m, i, s), the meaning of which will be given later in Section 3.1.

41


x

y

z

ϕ

θ r=ro

S

Figure 3.1: Electromagnetic structure embedded into a virtual sphericalmanifoldS of radiusr = r0.

In (3.1) we use the twist operator⊥n , defined in [11] for a general one-formA as

⊥nA = ⋆(n∧A), (3.2a)

where⋆ is the Hodge operator and∧ denotes the wedge product.A local picture of the twist operator acting on a general one-formA is

shown in Figure 3.2. The two-formn∧A is shown as an oriented surface.We can see also the tangential part of the one-formA with respect to thenormal directionn defined as

(A)t = ny(n∧A), (3.2b)

where the angle operatory denotes the contraction (for more details see[29]).

The normal one-formn is normal to the direction of a spherical man-ifold S (r) with a given radiusr and forms a rectangular frame with thetwo normal one-formsu andv, which are tangential to the surface of the

42

n

A

n∧A

(A)t = ny(n∧A)

⊥nA =⊥n (A)t = ⋆(n∧A)

Figure 3.2: A local picture of the twisted one-form⊥nA.

manifold. This is shown in Figure 3.3. The unit normal one-form n maybe written as

n = nxdx+nydy+nzdz= dr, (3.3a)

with

n2x+n2

y+n2z =

(

∂r∂x

)2

+

(

∂r∂y

)2

+

(

∂r∂z

)2

=

=

± x√

x2+y2+z2

2

+

± y√

x2+y2+z2

2

+

± z√

x2+y2+z2

2

=

= 1.

(3.3b)

We can see from (3.1) that the transverse basis one-forms of the mag-netic field intensity can be obtained from the transverse basis one-forms ofthe electric field intensity. The generalized voltages and currents for each

43


dx

dydz

S (r)

n

u

v

Figure 3.3: Spherical manifoldS (r) with the tangent framen, u andv.

modep may be obtained from the total field intensities by integration overthe spherical manifoldS (r) with radiusr by

Vp(r, t) ="

S (r)(−⊥nE(r,ϑ,ϕ, t))∧ ep(ϑ,ϕ), (3.4a)

Ip(r, t) ="

S (r)−H(r,ϑ,ϕ, t)∧ ep(ϑ,ϕ). (3.4b)

The generalized voltages are related to the generalized currents bygen-eralized impedances. In time-domain the relation is given by

Vp(r, t) = Zp(r, t) ∗ Ip(r, t), (3.5)

where∗ denotes the convolution in time. The generalized impedanceZp(r, t)describes completely the coupling between the electric andmagnetic fieldintensities. The generalized impedance isr dependent due to the varyingcross section of the half-space given byr ≥ r0 for a given radiusr = r0

Assuming harmonic excitation, the frequency domain representationsof the field quantities and the generalized network quantities can be used.

44

3.1 Radiation Modes

Using the phasor representations for the field quantities the relation (3.5)reads

Vp(r,ω) = Zp(r,ω)I p(r,ω). (3.6)

3.1 Radiation Modes

The transverse magnetic (TM) and transverse electric (TE) radiation modesare derived from the magnetic one-form potentialA and the electric one-form potentialF , respectively, which are oriented in the dr direction

A = Ar dr = rψdr, F = Fr dr = rψdr. (3.7)

In (3.7),ψ is a scalar function of spherical coordinates satisfying the scalarHelmholtz equation

(⋆d⋆d+k2)ψ(r,ϑ,ϕ) = 0 (3.8)

in spherical coordinate system, wherek = ω√εµ is the wavenumber,ω

the angular frequency andε andµ the permittivity and permeability of thebackground space, respectively.

The field intensities of the TM radiation modes are obtained from themagnetic vector potentialA by

H = ⋆dA, E = 1jωε

⋆d⋆dA. (3.9a)

The field intensities of the TE radiation modes are obtained from the elec-tric vector potentialF by

E = −⋆dF , H = 1jωµ

⋆d⋆dF . (3.9b)

The general solution of (3.8), obtained by separation of variables tech-nique, is given by

ψ(r,ϑ,ϕ) = bn(kr)Lmn (cosϑ)h(mϕ), (3.10)

45


wherebn(kr) are the spherical Bessel functions,Lmn (cosϑ) the associated

Legendre functions andh(mϕ) the harmonic functions. Due to the proper-ties of the functionsh(mϕ) andLm

n (cosϑ), and if we restrict our consider-ations to physically allowed solutions, we have to choosen andm integer,with n= 0,1,2, . . . and 0≤m≤ n.

We assume that all sources are confined within the spherical simula-tion region. Therefore, outside the sphere only outward propagating wavesoccur and the Sommerfeld radiation condition is fulfilled. Thus, we takebn(kr) = h(2)

n (kr) in (3.10), withh(2)n (kr) being the spherical Hankel func-

tions of the second kind.We definereal spherical harmonics Yinm(ϑ,ϕ) with n = 0,1,2, . . ., 0 ≤

m≤ n andi = eor o (even or odd) as

Yenm(ϑ,ϕ)

Yonm(ϑ,ϕ)

= γnmPmn (cosϑ)

cosmϕ

sinmϕ

, (3.11)

where

γnm=

√

ǫm2n+1

4π(n−m)!(n+m)!

(3.12)

with ǫm = 1 or 2 for m= 0 or m≥ 1, respectively, andPmn (cosϑ) being

associated Legendre polynomials [30]. Spherical harmonics satisfy the or-thonormality relations on a spherical surface, thus

∫ 2π

0

∫ π

0Yi

nm(ϑ,ϕ)Yi′n′m′ (ϑ,ϕ)sinϑdϑdϕ =

=

1 if n= n′, m=m′ = 0, i = i′ = e,

1 if n= n′, m=m′ > 0, i = i′,

0 otherwise.

(3.13)

To obtain the TM and TEradiation modeswe take

Ar = Fr = rYinm(ϑ,ϕ)h(2)

n (kr) (3.14)

in (3.7). The transverse basis one-formsep(ϑ,ϕ) are obtained from (3.9a)and (3.9b) by taking the radiusr and frequencyω independent directions

46

3.2 Cauer Realization of Radiation Modes

only. Since the magnetic one-form potentialA equals to the electric one-form potentialF , there is a simple relation between the TM and TE radia-tion modes. The transverse basis one-forms in terms of spherical harmon-ics are given by

eenmi(ϑ,ϕ) =

(

dϑ∂

∂ϑ+dϕ

∂

∂ϕ

)

Yinm(ϑ,ϕ)√

n(n+1), (3.15a)

ehnmi(ϑ,ϕ) =⊥n e

enmi, (3.15b)

where the superscriptse andh stand for TM(E) and TE(H) radiation modes,respectively.

We conclude that the transverse basis one-forms of the TM radiationmodes are given by

(

eenme

)

ϑ(ϑ,ϕ)(

eenmo

)

ϑ(ϑ,ϕ)

= cnm(−sinϑ)Pmn′(cosϑ)

cosmϕ

sinmϕ

(3.16a)

(

eenme

)

ϕ(ϑ,ϕ)(

eenmo

)

ϕ(ϑ,ϕ)

= cnmmPmn (cosϑ)

−sinmϕ

cosmϕ

(3.16b)

with cnm= γnm/√

n(n+1) being normalization coefficients which dependonn andm. The prime′ denotes derivation of a function with respect to itsargument.

The transverse basis one-forms of the TE radiation modes areobtainedfrom (3.16) using (3.15b).


Let us assume the complete electromagnetic structure underconsiderationembedded in a virtual sphereS as shown in Figure 3.1.

The wave impedances of the TM radiation modes on the surface of thesphereS are given by

Zenm= jη

H(2)n′(kr0)

H(2)n (kr0)

(3.17)

47


and of the TE radiation modes by

Zhnm= − jη

H(2)n (kr0)

H(2)n′(kr0)

, (3.18)

whereη =√

µ/ε is the wave impedance of the plane wave andH(2)n (kr) =

krh(2)n (kr). Please note that the characteristic wave impedances depend only

on the indexn for the given radiusr0 of the virtual sphere.Using the recurrence formula for the spherical Hankel functions [30], the

Cauer representations of the TM and TE radiation modes may beobtained[10], [31]. The impedance of the TM modes is then written as

Zenm= η

njkr +

1

2n−1jkr +

12n−3

jkr +

. . .

+1

3jkr +

11jkr +1

, (3.19a)

and of the TE modes as

Zhnm= η

1

njkr +

1

2n−1jkr +

12n−3jkr +

. . .

+1

3jkr +

11jkr +1

. (3.19b)

The equivalent Cauer circuits realizing the impedances of the radiationmodes given by (3.19) are shown in Figure 3.4 and Figure 3.5.

48


µ

ε r2n−3

ε rn

r2n−1 2n−5

µr ηZen

Figure 3.4: Cauer realization of the impedance of TMn radiation mode.

µ

r2n−5

µrn

r

ε r2n−1

2n−3

ε

ηZhn

Figure 3.5: Cauer realization of the impedance of TEn radiation mode.

In Figure 3.6 – Figure 3.9 the plots of real and imaginary parts of theimpedances of radiation modes realized by the Cauer circuits and com-puted analytical by using (3.17) and (3.18) are compared. The plots areshown for both, the TM and TE radiation modes in free-space (η = 377Ω),and mode numbersn= 1,2,3,4,5,25,30.

49

3M

ultipoleE

xpansion

0.1 1 10 100kr [-]

0

100

200

300

377

Re

Ze n

[Ω]

n=1

n=2

n=3

n=4

n=5

n=25 n=30

TMn - Cauer circuits

TMn - Analytical expressions

Fig

ure

3.6

:TM

radiatio

nm

od

es–

realparto

fthe

imp

edan

ce.

50

3.2C

auerR

ealizationofR

adiationM

odes

0.1 1 10 100kr [-]

-1000

-900

-800

-700

-600

-500

-400

-300

-200

-100

0

ImZ

e n [Ω

]

n=1

n=2

n=3

n=4

n=5

n=25 n=30

TMn - Cauer circuits

TMn - Analytical expressions

Fig

ure

3.7

:TM

radiatio

nm

od

es–

imag

inary

parto

fthe

imp

eda

nce.5

1

3M

ultipoleE

xpansion

0.1 1 10 100kr [-]

0

200

600

800

1000

1200

377

Re

Zh n

[Ω]

n=1

n=2

n=3

n=4

n=5

n=25 n=30

TEn - Cauer circuits

TEn - Analytical expressions

Fig

ure

3.8

:TE

radiatio

nm

od

es–

realparto

fthe

imp

edan

ce.

52

3.2C

auerR

ealizationofR

adiationM

odes

0.1 1 10 100kr [-]

0

200

400

600

800

1000

1200

ImZ

h n [Ω

]

n=1 n=

2

n=3 n=

4 n=5

n=25 n=30

TEn - Cauer circuits

TEn - Analytical expressions

Fig

ure

3.9

:TE

radiatio

nm

od

es–

imag

inary

parto

fthe

imp

eda

nce.5

3

4 Connection Subnetwork

4.1 Tellegen’s Theorem and ConnectionSubnetwork

Tellegen’s theorem is a fundamental theorem in network theory whichhas been given by Tellegen [32] and generalized by Penfield, Spence andDuinker [33]. Notably, the form of the theorem is simple and general. Thebeauty and power of the theorem is in it’s simplicity and generality.

The importance of the theorem follows from the fact that it’sproof de-pends solely upon Kirchhoff’s laws. Consequently, the theorem is validfor all types of networks which satisfy Kirchhoff’s laws regardless if theseare linear or non-linear, time invariant or time-varying, reciprocal or non-reciprocal, passive or active. The condition which must be satisfied for thetheorem to be valid is that the circuits under considerationhave commontopology.

Since the Tellegen’s theorem plays an important role also inthe network-oriented modeling, the network form of the theorem will be now given.Then, the field form of the Tellegen’s theorem and its discretized form asgiven in [1] will be described.

4.1.1 Network form of Tellegen’s theorem

The Tellegen’s theorem states that for two networks sharingthe same topol-ogy and havingB branches the following equation holds

B∑

b=1

i′b v′′b = 0, (4.1)

55


wherei′b are branch currents of one network andv′′b are branch voltages ofthe other network.

The power and usefulness of the theorem can be seen from the two fol-lowing conclusions, which follow directly from the theorem.

1. If the two networks considered in the formulation of the Tellegen’stheorem are the same then (4.1) can be physically interpreted as theconservation of energy principle within the network and hasthe form

B∑

b=1

i′bv′b = 0. (4.2)

2. With the previous interpretation of the Tellegen’s theorem as energyconservation we can conclude from (4.1) that there does not existany network which would not conserve energy. In other words,allnetworks obeying Kirchhoff’s laws must conserve energy.

In the generalized form [33] the Tellegen’s theorem has the form

B∑

b=1

Λ′(i′b) Λ′′(v′′b ) = 0, (4.3)

whereΛ′ andΛ′′ are Kirchhoff current and voltage operators, respectively.Kirchhoff operators, as introduced in [33], are such operators which trans-form given electrical quantities obeying Kirchhoff’s laws into new electri-cal quantities obeying also Kirchhoff’s laws.

Examples of allowed Kirchhoff operators are

• differentiation with respect to time,

• multiplication by a constant,

• any scalar linear transformation.

The proof of the Tellegen’s theorem and it’s generalized form may befound in [33]. A physical approach to the proof has been givenby Temesin [34].

56

4.1 Tellegen’s Theorem and Connection Subnetwork

If the setsi′b andv′′b are interpreted as vectorsi′ = i′b andv′′ = v′′b then (4.1) can be written as follows

i′Tv′′ = v′′T i′ = 0. (4.4)

This equation can be interpreted as orthogonality relations, i.e., the vectorsi′ andv′′ are mutually orthogonal.

4.1.2 Field form of Tellegen’s theorem

The field form of Tellegen’s theorem states that [1], [11]∮

∂VE′(x, t′)∧H ′′(x, t′′) = 0, (4.5)

where′ and′′ denote independent field forms defined on a common mani-fold ∂V. The theorem states that no energy can be stored inside the mani-fold ∂V of zero measure.

4.1.3 Connection subnetwork

Scattering matrix of ideal transformer network

The equations of a transformer network as shown in Figure 4.1, with equalreference resistances at the ports, can be summarized in theequation

[

I −NNT I

]

b =[

−I NNT I

]

a (4.6)

with

N =

n11 n12 . . . n1N

n21 n22 . . . n2N

. . . . . . . . . . . . . . . . . . . . . .

nM1 nM2 . . . nMN

(4.7)

summarizing the transformer ratios and

a=[

aN+1 aN+2 . . . aN+M a1 a2 . . . aN

]

T , (4.8a)

b =[

bN+1 bN+2 . . . bN+M b1 b2 . . . bN

]

T (4.8b)

57


1:n11 1:n21 1:nM1

1:n12 1:n22 1:nM2

1:n1N 1:n2N 1:nMN

(a1,b1)

(a2,b2)

(aN,bN)

(aN+1,bN+1) (aN+M,bN+M)

Figure 4.1: Connection subnetwork consisting of ideal transformers.

58

4.1 Tellegen’s Theorem and Connection Subnetwork

being the reorganized vectors of incoming and reflected waves.The reorganized scattering matrixS is computed as

S=[

I −NNT I

]−1 [

−I NNT I

]

=

[

S4 S3

S2 S1

]

, (4.9)

where the inverse matrix can be obtained using the techniqueof Schurcomplement(see Section A.3.2).

The scattering matrixS is related to the reorganized scattering matrixSas follows

S=[

S1 S2

S3 S4

]

. (4.10)

The resulting scattering matrix of the connection subnetwork is given as

S=[

I −2NTS−1A N NTS−1

A +NT −NTS−1A NNT

2S−1A N −S−1

A +S−1A NNT

]

, (4.11)

withSA = I +NNT , (4.12)

whereSA is the Schur complement.

Examples

In the following let us examine some special cases of the scattering matrix(4.11).

Orthonormal square N

In this caseNNT = NTN = I (4.13)

and

SA = 2I ; S−1A =

12

I . (4.14)

The (4.11) is simplified to

S=[

0 NT

N 0

]

(4.15)

59


NT is a right inverse of N

In this caseNNT = I (4.16)

butNTN , I (4.17)

The

SA = 2I ; S−1A =

12

I . (4.18)

The (4.11) is simplified to

S=[

I −NTN NT

N 0

]

(4.19)

Ideal transformer: N-ports primary, one-port secondary

The transformer ratios are denoted byni with i = 1, . . . ,N. The matrixN isreduced to a vector

N =[

n1 n2 . . . nN

]

(4.20)

The (N+1)× (N+1) S-matrix is given by

S= D

1D +n2

1 n1n2 n1n3 . . . n1nN −n1

n2n11D +n2

2 n2n3 . . . n2nN −n2...

......

......

...

nNn1 nNn2 nNn3 . . . 1D +n2

N −nN

−n1 −n2 −n3 . . . −nN1D +1

(4.21a)

with

D = − 2

1+n21+ . . .+n2

N

. (4.21b)

Note that ifn21+ . . .+n2

N = 1 thenD = −1 and the scattering matrix has azero at the last element.

60

4.2 Scattering Matrix of the Connection Subnetwork

For the special case ofN = 1, the scattering matrix computed from (4.21)is reduced to

S=1

1+n21

[

1−n21 2n1

2n1 n21−1

]

, (4.22)

which may be found in the literature (e.g., see [35] p.41).

4.2 Scattering Matrix of the ConnectionSubnetwork

4.2.1 Direct derivation

Boundary conditions

In the following, the TLM simulation domain (R1) will be referred to assubdomain 1 and the outside space (R2) as subdomain 2. The two sub-domains are identified by a spherical surface (S ) between the two (seealso Figure 1.1). Also, it will be convenient to use the vector notation forelectromagnetic fields in this Section.

First, the tangential electric field on the interfacing surface (i.e. on thesphere) is expanded into two sets of vector basis functionse(1)

n ande(2)m with

indicesn= 1,2, . . . ,N andm= 1,2, . . . ,M. The electric fields can be writtenas

E(1)t =

N∑

n

V(1)n e(1)

n , (4.23a)

E(2)t =

M∑

m

V(2)m e(2)

m , (4.23b)

with superscripts(1) and(2) denoting the subdomain 1 and 2, respectively.We call the modal amplitudesV(1)

n andV(2)m equivalent voltages.

Considering equality of the tangential fields represented by (4.23) wecan write

E(1)t = E(2)

t . (4.24)

61


Multiplying both sides of the equation withe(2) and integrating overS weget

∫

S

N∑

n

V(1)n e(1)

n ·e(2)m dS =

∫

S

M∑

m′V(2)

m′ e(2)m′ ·e

(2)m dS , (4.25a)

whereS denotes the domain of integration, here the spherical surface be-tween subdomain 1 and 2. Assuming orthonormality of the basis func-tions (4.25a) results in

V(2)m =

N∑

n

V(1)n

∫

S

e(1)n ·e(2)

m dS . (4.25b)

The tangential magnetic fields in the subdomain 2 are relatedto the tan-gential electric fields in the same subdomain via

H(2)t =

M∑

m

I (2)m r0×e(2)

m , (4.26a)

with

I (2)m =

V(2)m

Z(2)m

= Y(2)m V(2)

m , (4.26b)

whereZ(2)m andY(2)

m areequivalent impedancesandequivalent admittancesof them-th radiation mode. The quantitiesI (2)

m andI (1)n are calledequivalent

currents.Similar considerations may be done for the tangential magnetic fields

resulting in an equation analogous to (4.25b). The equationrelates theequivalent currents in domain 1 to those in domain 2 and reads

I (1)n =

(

−1)

M∑

m

I (2)m

∫

S

e(2)m ·e(1)

n dS . (4.27)

62


Next, we define the coefficientnmn by

nmn=

∫

S

e(2)m ·e(1)

n dS (4.28)

and the matrixN as

N =

n11 n12 . . . n1N

n21 n22 . . . n2N

. . . . . . . . . . . . . . . . . . . . . .

nM1 nM2 . . . nMN

. (4.29)

Mapping between network quantities and wave quantities

The equivalent voltages and equivalent currents in subdomain 1 are trans-formed to the wave quantities by

V(1)n = a(1)

n +b(1)n , (4.30a)

ηI (1)n = a(1)

n −b(1)n , (4.30b)

wherea(1)n andb(1)

n are the amplitudes of the incoming and outgoing waveswith respect to the surface of the sphere and whereη =

√

µ/ε.

On the boundary of subdomain 2 we introduce local incoming wavesa(2)m

and local reflected wavesb(2)m in a similar manner. The equivalent voltages

and currents are related to the wave quantities by

V(2)m = a(2)

m +b(2)m , (4.31a)

ηI (2)m = a(2)

m −b(2)m . (4.31b)

The Scattering Matrix

Inserting (4.30) and (4.31) into (4.25b) and (4.27) and with(4.28) we ob-tain the equations of the connection subnetwork between thetwo subdo-

63


mains

a(2)m =

N∑

n

(

a(1)n +b(1)

n

)

nmn

−b(2)m , (4.32a)

a(1)n = b(1)

n −

M∑

m

(

a(2)m −b(2)

m

)

nmn

. (4.32b)

Equations (4.32) can be rewritten as

b(2)m −

N∑

n

b(1)n nmn= −a(2)

m +

N∑

n

a(1)n nmn, (4.33a)

b(1)n +

M∑

m

b(2)m nmn= a(1)

n +

M∑

m

a(2)m nmn. (4.33b)

Using matrix notation, the left hand side of equations (4.33) can be writtenas

−n11 −n12 . . . −n1N 1 0 . . . 0−n21 −n22 . . . −n2N 0 1 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−nM1 −nM2 . . . −nMN 0 0 . . . 11 0 . . . 0 n11 n21 . . . nM1

0 1 . . . 0 n12 n22 . . . nM2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 . . . 1 n1N n2N . . . nMN

b(1)1

b(1)2...

b(1)N

b(2)1

b(2)2...

b(2)M

, (4.34a)

which can be written in short form using block matrixB as

[

−N II NT

]

b = Bb, (4.34b)

whereb is the vector of outgoing waves.

64


In a similar way we can write the right hand side of equations (4.33) as

n11 n12 . . . n1N −1 0 . . . 0n21 n22 . . . n2N 0 −1 . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

nM1 nM2 . . . nMN 0 0 . . . −11 0 . . . 0 n11 n21 . . . nM1

0 1 . . . 0 n12 n22 . . . nM2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 . . . 1 n1N n2N . . . nMN

a(1)1

a(1)2...

a(1)N

a(2)1

a(2)2...

a(2)M

, (4.35a)

which can be written in short form using block matrixA as[

N −II NT

]

a= Aa, (4.35b)

wherea is the vector of incoming waves.Equation (4.34b) must be equal to (4.35b) by (4.33). Taking the inverse

matrixB−1 we obtainb = B−1Aa = Sa, (4.36)

whereS is the scattering matrix of the connection between subdomain 1and 2.

Reorganizing the vectors of incoming and outgoing waves according to

a=[

a(2)1 a(2)

2 . . . a(2)M a(1)

1 a(1)2 . . . a(1)

N

]

T , (4.37a)

b =[

b(2)1 b(2)

2 . . . b(2)M b(1)

1 b(1)2 . . . b(1)

N

]

T , (4.37b)

equations (4.34b) and (4.35b) take the same form as the equation (4.6).This shows that the transformation ratios of the connectionsubnetwork in(4.7) are to be taken according to (4.28).

Discretized electromagnetic field

When the electromagnetic field functions are discretized inspace, the equal-ity (4.24) cannot be satisfied, unless the basis functions inboth subdomains

65


form a complete set and the summations are taken over all basis functions.However, pointwise equality for the sampled field functionscan be satis-fied.

Consider a set of pointsRS , elements of which are all discrete pointson the surface of the spherical domain, i.e.,r s ∈ RS . Index s denotes aparticular point on the surface. Using the sampling property of the Diracdelta function we can write

E(1)(r s) =√

∆A(r s)∫

S

E(1)t (r )δ(r − r s)dS , (4.38)

where we leave out the subscriptt for discretized tangential fields andwhere

∆A(r s) = ∆As =

∫

V (r s)dS . (4.39)

The domainV (r s) is the Voronoi region around pointr s. Furthermore,

∫

S

dS ≈S

∑

s

∆As= 4π. (4.40)

WithEs= E(r s) (4.41)

we can define the vector

EEE =

E1,E2, . . . ,ES

T . (4.42)

The equality (4.24) can be satisfied now and reads

EEE(1) =EEE(2). (4.43)

4.2.2 Implementation Issues

The number of elements to be stored for any S-matrix of the permissibleconnection network:

elements=N(N+1)

2, (4.44)

66


since the matrix is symmetric.The memory requirements to store the scattering matrix are following:

memory=648

11024×1024×1024

×elements≈ 7.45×10−9× N(N+1)2

[GB].

(4.45)A 10000×10000 matrix requires cca 382MB of memory.

67

5 Transmission Line Matrix –Multipole Expansion Method

In this Chapter we use the results obtained in previous chapters as a basisfor the hybrid Transmission Line Matrix – Multipole Expansion (TLMME)method. Individual parts of the hybrid TLMME method will be describedin detail.

We begin with the description of a spherical TLM region. Then, the pre-processing part of the TLMME method is given. During pre-processingthe spherical TLM region is generated, a scattering matrix of the connec-tion subnetwork is computed and Wave Digital Filter (WDF) models ofimpedances of radiation modes are created. We will see how these filtersare implemented and synchronized with the TLM method.

The processes of generation of the spherical TLM region, meshing ofgeometric objects and creation of recursive WDF models of impedancesof radiation modes have been fully automatized. This avoidserrors intro-duced by manual pre-processing and enables fast pre-processing of techni-cally interesting models.

The Chapter is concluded with the description of the TLMME algorithm.

5.1 Spherical TLM Region

Since the transverse basis one-formsep(ϑ,ϕ) in (3.15) are defined on thesurface of a spherical manifold, we cannot simply proceed with the con-nection of the classical cubical TLM simulation domain to the radiationmodes.

Therefore, we embed the TLM simulation domain into a sphere,whichis discretized in a rectangular mesh, and obtain a sphericalTLM region

69

5 Transmission Line Matrix – Multipole Expansion Method

Figure 5.1: The spherical TLM region – discretized spherical manifoldS .

bounded by a discretized spherical manifoldS as depicted in Figure 5.1.Inside the spherical TLM region the electromagnetic field ismodeled

by TLM. On the boundary of the spherical domain – on the discretizedspherical manifoldS – we define a set of surfaces on which the fieldsresulting from the TLM algorithm are obtained. This set willbe denotedby S

C and its elements (surfaces) bysc ∈SC. The fields onsc are used

to compute the modal coefficients in (5.2).

5.1.1 Connection of the simulation domain to theradiation modes

The transverse field one-forms are expressed as a linear combinations ofthe basis one-forms by

Nt(r,ϑ,ϕ, t)∣

∣

∣

∣

r=r0=

∑

n,m,i,s

αsnmi(t)s

snmi(ϑ,ϕ), (5.1)

where the indexs= eor h stands for TM(E) or TE(H) modes, respectively,ands stands for the basis structure one-formse or h as defined in (3.1). In(5.1)N stands for the one-form fieldsE orH .

70

5.1 Spherical TLM Region

Using the orthonormal property of the structure one-forms,the time-dependent modal coefficientsαs

nmi(t) are given by

αsnmi(t) =

∫ 2π

0

∫ π

0

(

−⊥nN(r,ϑ,ϕ, t)∣

∣

∣

∣

r=r0

)

∧ ssnmi(ϑ,ϕ). (5.2)

For simplicity of the notation, let us consider in the following just onemode with givenn,m, i, s.

We may write the structure one-forms in (5.2) as

s = sϑmϑdϑ+ sϕmϕdϕ, (5.3)

with mϑ andmϕ being the metric coefficients of the spherical manifold.On each surfacesc of the discretized manifold ofS (r) we have knowl-

edge ofNx(x,y,z, t), Ny(x,y,z, t) or Nz(x,y,z, t). After projecting these fieldvalues onto the spherical manifoldS (r), we can write (5.2) as

α(t) =∫ 2π

0

∫ π

0

Nx(ϑ,ϕ, t)[

t12sϑmϑ + t13sϕmϕ

]

+

+Ny(ϑ,ϕ, t)[


]

+

+Nz(ϑ,ϕ, t)[


]

dϑ∧dϕ,

(5.4)

with t12 = ∂x/∂ϑ, t13 = ∂x/∂ϕ, t22 = ∂y/∂ϑ, t23 = ∂y/∂ϕ, t32 = ∂z/∂ϑ andt33 = ∂z/∂ϕ . For the considered spherical manifold the metric coefficientsaremϑ = 1 andmϕ = sinϑ.

Using the point-matching technique, the time-dependent modal coeffi-cients can be computed numerically by

α(t) =∫ 2π

0

∫ π

0

[

S Px+S Py+S Pz]

sinϑdϑdϕ =∫ 2π

0

∫ π

0S Psinϑdϑdϕ,

(5.5)with

S Px = Nx(ϑ,ϕ, t)[

t12sϑ + t13sϕ]

, (5.6a)

S Py = Ny(ϑ,ϕ, t)[

t22sϑ + t23sϕ]

, (5.6b)

S Pz= Nz(ϑ,ϕ, t)[

t32sϑ + t33sϕ]

. (5.6c)

71


The integral in (5.5) is obtained numerically as

α(t) ≈∑

ϑd

∑

ϕd

S P(ϑd,ϕd, t)sinϑd∆ϑd ∆ϕd , (5.7)

with∑

ϑd

∆ϑd = π,∑

ϕd

∆ϕd = 2π. (5.8)

In (5.7),ϑd andϕd are the discrete elevation and azimuth, respectively, atthe points where the fields of the surface elementsc are defined, and∆ϑd

and∆ϕd are the spatial discretization steps in the corresponding direction,which differ with the location of the elementsc.

The modal coefficients in (5.7) represent the input of the correspondingradiation mode. Each TLM port of the surface elementsc contributes tothe total radiation. The lumped element equivalent circuitmodel of theconnection circuit connecting the TLM network representing the simula-tion domain with the ladder networks representing the radiation modes isshown in Figure 5.2. In [1] it has been shown, that the connection networkcontains only ideal transformers.

5.2 Pre-processing

Pre-processing for the TLMME method consists of the following steps:

1. Generation of the spherical TLM region,

2. computation of the scattering matrix of the connection subnetwork,

3. creation of the WDF models of radiation modes,

4. meshing of the physical objects in the TLM method,

5. generation of the models of excitation.

The generation of the spherical TLM region is done automatically usinga Cartesian mesh generator which is described in the next Section.

72

5.2 Pre-processing

a11

a12

a1k1

b11

b12

b1k1

a21

a22

a2k2

b21

b22

b2k2

am1

am2

amkm

bm1

bm2

bmkm

η

1:n11

1:n12

1:n1k1

1:n21

1:n22

1:n2k2

1:nm1

1:nm2

1:nmkm

n-th radiation mode

Figure 5.2: Network model of the connection of one radiationmode to theTLM simulation domain.

73


WDF models of the impedances of radiation modes are also generatedautomatically by taking advantage of their recursive structure. More detailsare given in Section 5.2.2.

5.2.1 Cartesian mesh generation

The discretization procedure we are using here is an extension of the methoddescribed in [36] for 2D manifolds. The discretization of 2Dclosed mani-folds follows these steps:

1. create a model of the 2D manifold,

2. define the mesh,

3. discretize the object usingray shootingtechnique,

4. extract the boundary from the volumetric object.

This method works for volumetric objects which exhibit a closed bound-ary. However, using boolean operations the method may be extended todiscretize non-closed general 2D objects.

The principles of the ray shooting technique are shown in Figure 5.3. Aray with two constant spatial coordinates is examined for the intersectionswith the object under consideration. Since we know that the object has aclosed boundary, i.e., it is a finite volume object, we identify the objectin the prescribed mesh. Now, the boundary of the discretizedvolumetricobject may be easily extracted. As a result we get the 2D discretized man-ifold. This manifold may be represented by a set of elementary surfaces,e.g., byS C defined in Section 5.1.

Example: Cartesian meshing of a rectangular cavity resonat or

The 3D Transmission Line Matrix (TLM) method has been originally de-veloped in a structured Cartesian mesh [13], [26], [11]. Theformulationof the TLM algorithm in structured Cartesian mesh is the easiest one andthe implementation is straightforward. However, the lack of conformity of

74

5.2 Pre-processing

Figure 5.3: The ray shooting technique.

the Cartesian mesh to the boundaries of complex curved objects has beensoon recognized.

Researchers have tried to improve the conformity of the meshto theboundaries of the objects by using graded, conformal, unstructured triangu-lar and tetrahedral meshes [37]. However, all of these approaches increasethe complexity of the TLM computation. Higher memory requirements,mesh queries and additional specialized techniques increase the complex-ity of the TLM algorithm and computational time.

Another very important aspect is the possibility of parallelization of theTLM solver. Structured solvers are much easier to be parallelized thenunstructured solvers. In general, properly designed structured solvers arealso always faster and more robust than properly designed unstructuredsolvers.

Obviously, the main arguments why to use unstructured solvers is theconformity of the mesh to the object and easier mesh generation than fora structured solver. On the other hand, if we are able to approximate com-plex curved geometries within a high-resolution structured Cartesian mesh,i.e., a mesh where the spatial resolution is much higher thanthe curvature

75


of the object, the computed solution will converge to the solution com-puted with an unstructured mesh. A study on the influence of the stepped(staircase) boundaries has been done for the finite-difference time-domainmethod in [38].

Since TLM in the formulation of Symmetrical Condensed Node (SCN)(and all the other condensed nodes) has both the electric field componentsand the magnetic field components defined at the same discretespatial co-ordinate, it is easy to introduce boundary conditions in theface of the TLMcell. We may use the surfaces available from the TLM mesh to approximatethe curved two-dimensional objects. If the structured meshhas a sufficientresolution, the curvature of the objects is well approximated.

In this example we use a regular structured Cartesian mesh with a highspatial resolution – a High-Resolution Mesh (HRM). An HRM iscapa-ble of approximating curved objects with high accuracy, since the spatialdiscretization step is much smaller then the curvature of the object. Therectangular waveguide is discretized within this mesh. Dueto the relativeease of parallelization a parallel computational environment is also used.

An off-grid perfect boundary condition for the FDTD method has beenpublished by Rickard and Nikolova [39], implementing an enhanced stair-case approximations. We use the same rectangular resonatoras in theabove mentioned paper and compare the results obtained in the HRM.

The rectangular cavity resonator has dimensionsa= 10 mm,b= 20 mm,l = 30 mm, with thel-th dimension located along thez axis. The resonatoris rotated in theϑ andϕ directions and discretized using the HRM.

The configuration of the simulation and the simulation timesare sum-marized in Table 5.1 and Table 5.2, respectively. The calculated resonantfrequencies and the relative error are summarized in Table 5.3. For thesimulations computing nodes with Pentium4 3.0 GHz processors, 1GB 400DDR memory, connected with 1-Gbit/s switched Ethernet network, havebeen used.

5.2.2 Wave Digital Filter models of radiation modes

We have seen in Section 3.2 the equivalent Cauer canonical representa-tions of impedances of TE and TM radiation modes. In this Section these

76

5.2 Pre-processing

ϑ ϕ ∆l # Cells0 0 1 mm 28×18×38= 1915245 0 1 mm 40×18×40= 2880045 0 0.5 mm 76×32×76= 18483245 0 0.25 mm 156×52×156 1.265×106

45 45 0.25 mm 156×148×156 3.4×106

45 45 0.125 mm 300×284×284 24.2×106

Table 5.1: Configuration of the rectangular cavity resonator. The distribu-tion ratio in the distributed case is given in the brackets (x : y : z).

ϑ ϕ ∆l [mm] t standalone t distributed0 0 1 mm 1.43 min -45 0 1 mm 2.03 min -45 0 0.5 mm 9.72 min -45 0 0.25 mm 2.5 h 40.43 min; (2:3:1)45 45 0.25 mm 6.5 h 1.82 h; (2:3:1)45 45 0.125 mm - 21.78 h; (7:1:1)

Table 5.2: Simulation times of the rectangular cavity resonator.

ϑ ϕ ∆l [mm] f011; δ[%] f012; δ[%] f101; δ[%]0 0 1 9.0101; 0.027 12.493; 0.012 15.805; 0.02945 0 1 8.8223; 2.057 12.228; 2.107 15.556; 1.54745 0 0.5 8.9624; 0.5 12.45; 0.331 15.705; 0.60445 0 0.25 9.027; 0.215 12.537; 0.365 15.77; 0.19245 45 0.25 8.973; 0.385 12.434; 0.4599 15.717; 0.52845 45 0.125 9.0002; 0.082 12.479; 0.099 15.7811; 0.122

Table 5.3: Calculated resonant frequencies of the rectangular cavity res-onator. All frequencies are given in GHz. Next to the resonantfrequency the relative error is given.

77


Figure 5.4: Rectangular cavity resonator discretized withHRM; ϑ = 45,ϕ = 45, ∆l = 0.125 mm.

impedance representations will be transformed into their equivalent digitalcounterparts using Wave Digital Filters (WDFs) [25].

One main advantage of using WDFs over discrete filter structures ob-tained via other transformation techniques like direct bilinear transform orimpulse invariance is their usage of wave quantities as portsignals. Thisis a considerable advantage for the TLM method which is also based onwave quantities.

With WDFs we can directly realize in digital domains the digital coun-terparts of analog circuit elements like capacitors, inductors, resistors andinterconnections. Also, it is not difficult to generate ladder structures recur-sively. Since the equivalent representations of radiationmodes are ladderstructures, they can be efficiently generated using these techniques.

Capacitors, inductors, resistors and interconnections as WDFs

The behaviour of capacitors, inductors, resistors and interconnections canbe efficiently and elegantly transformed to digital domain with WDFs (see[25], pp. 274–280).

A capacitor is implemented as a time-delay by one time step with refer-

78

5.2 Pre-processing

ence port resistanceRC. The port resistanceRC is given by

RC =1C∆t2, (5.9)

whereC is the capacitance of the analog capacitor to be modeled and∆t isthe discrete time step of the digital system. For our purposes∆t is equal tothe time step of the TLM algorithm.

Similarly, an inductor is implemented as a time-delay by onetime stepwith reference port resistanceRL. The port resistanceRL is given by

RL = L2∆t, (5.10)

whereL is the inductance of the analog inductor to be modeled.Notably, the WDF realizations of capacitors and inductors lead to the

same results as those obtained from transmission line considerations byChristopoulos (see [26], pp. 25–38).

The port resistance of a resistor is chosen to equal the resistance of theanalog resistor. Consequently, a resistor can be easily modeled as an ele-ment without any reflection.

For the modeling of interconnection networks we use the parallel andseries adaptors. The port resistances of the interconnection network can bechosen independently. The scattering matrices of the adaptors are delay-free and energy conservative.

Recursive generation of ladder structures

For the recursive generation of WDF ladder structures therealizabilitycon-dition is important. Realizability is granted when no delay-free directedloop exists in the signal-flow diagram of the WDF and if the total delay inany loop is equal to a positive integer multiple or zero multiple of the timestep∆t (see [25], pp. 271–272).

To assure a realizable WDF ladder structure we can use constrained in-terconnection networks. A constrained adaptor has one input port matched.Consequently, no reflections occur at this port and the outgoing wave ofthe constrained port is independent of the incoming wave on that port. To

79


match the constrained port we need to chose appropriately the referenceimpedance of that port.

An example of a WDF realization of the impedance of TM3 radiationmode is shown in Figure 5.5. Note the constrained interconnection net-

T

RC1

R0

R'1

T-1

RL1

T

RC2

R'2

R'3

T-1

RL2

η

a = 0

Figure 5.5: WDF ladder structure implementing the impedance of TM3 ra-diation mode.

works. The constrained port is denoted by a short bar at the signal path ofthe outgoing wave. From the figure we can see the periodic structure ofthe digital filter, where the models for capacitors and inductors are alter-nating. This periodicity is implemented efficiently by means of a recursivealgorithm.

5.3 The TLMME Algorithm

The difference between TLM and TLMME algorithms is in the applicationof boundary conditions. From the point of view of the TLM method theTLMME algorithm looks like a TLM algorithm with a special boundarycondition1.

The discrete time evolution is implemented in the TLM methodby meansof iterations. One iteration corresponds to the time evolution by one timestep. The iterations are repeated for as many time steps as required. OneTLM iteration is shown in Figure 5.6. We can see that the TLM itera-

1This is a result which we expect. Eventually, the ME is simulating a radiation boundarycondition.

80


Scatter

Apply boundary conditions

Connect

Figure 5.6: One TLM iteration – evolution by one time step.

tion begins with the scattering process – theScatterprocedure. The scatterprocedure is applied to each TLM cell.

Next, the boundary conditions are processed – theApply boundary con-ditions procedure. At this state of the iteration the wave quantities arelocated at the interfaces of the TLM cells. Inside this procedure the excita-tion is processed and the boundary conditions are applied.

Eventually, the wave quantities between neighbouring cells are exchanged– theConnectprocedure – and become incoming waves for the next itera-tion. The iteration can start from the beginning.

The TLMME algorithm is implemented inside the Apply boundary con-ditions procedure and is schematically depicted in Figure 5.7.

In the first step we need to store the values of outgoing waves on the dis-cretized spherical manifoldS (see Figure 5.1) and outgoing waves fromthe WDF models of radiation modes – theGet values on sphereandGetoutgoing WDFprocedures.

We can summarize these waves in the vectoraT LMME defined as

aT LMME =

[

aT LM

aME

]

, (5.11)

where the vectorsaT LM andaME denote the outgoing waves of the spher-

81


Get values on sphere

Multiply with connection subnetwork matrix

Iterate over WDF models Set values on sphere

Get outgoing WDF

Save WDF state

Figure 5.7: One TLMME iteration – ME boundary condition.

ical manifold and the outgoing waves from the WDF models of radiationmodes, respectively. VectoraT LMME represents the incoming waves on theconnection subnetwork.

Next, we compute the outgoing waves of the connection subnetwork –theMultiply with connection subnetwork matrixprocedure. The vector ofoutgoing waves from the connection subnetwork is denoted bybT LMME,

bT LMME =

[

bT LM

bME

]

. (5.12)

The result of the Multiply with connection subnetwork matrix procedure isthe vectorbT LMME calculated as

bT LMME = ST LMMEaT LMME, (5.13)

whereST LMME is the scattering matrix of the connection subnetwork. Thereis no time delay present in this scattering process.

82


The incoming waves onto the spherical manifold of the TLM domainare now available in the vectorbT LM and the incoming waves onto theWDF models of the radiation modes in the vectorbME. The incomingwavesbT LM are set on the spherical manifold in theSet values on sphereprocedure. The incoming wavesbME are used for one time step iterationover all WDF models – theIterate over WDF modelsprocedure.

Eventually, the state of the WDF models needs to be stored in order tobe used during the next TLMME iteration. This is done in theSave WDFstateprocedure.

83

6 Numerical Examples

6.1 Flat Dipole Antenna

The first example on which the TLMME method and it’s features will bedemonstrated is a flat dipole antenna. This example is simpleenough to besolved to some extent analytically, but does include enoughcomplexity tohighlight important behaviour of the TLMME method. The layout of theflat dipole antenna is shown in Figure 6.1.

302

1 30

PEC

Figure 6.1: Top view of the flat dipole antenna; all lengths are given in mmunits.

To evaluate the performance of the TLMME method, the input admit-tance of the flat dipole is chosen as the key antenna characteristics. Itis well known and generally valid that the input admittance/impedance ismuch more sensitive to the accuracy of the modelling method than otherantenna parameters like the radiation pattern (cf., e.g., [40]).

In this example the input admittance of the flat dipole is firstcalculatedanalytically. Then, a TLM computation with absorbing boundary condi-tions on cubical simulation region and a TLMME computation results willbe presented.

85


6.1.1 Analytical characterization

For the analytical characterization of the flat dipole antenna an approxima-tion by a thin wire dipole antenna is done (see [40], p. 454). The equivalentradiusae of the wire dipole is given by

ae= 0.25w, (6.1)

wherew = 2mm is the width of the flat dipole antenna (see Figure 6.1).Using the induced emf method (see [41], pp.359–434) the realand imagi-nary parts of the input impedance referred to at the current maximum aregiven by ([40], p. 410)

Rm=Z0

2π

[

C+ ln(kl)−Ci(kl)+12

sin(kl)(

Si(2kl)−2Si(kl))

+12

cos(kl)

(

C+ ln

(

kl2

)

+Ci(2kl)−2Ci(kl)

)]

,

(6.2a)

Xm=Z0

4π

[

2Si(kl)+cos(kl)(

2Si(kl)−Si(2kl))

−sin(kl)

(

2Ci(kl)−Ci(2kl)−Ci

(

2ka2e

l

))]

,

(6.2b)

whereZ0 = 377,k = ωc0

, C = 0.577215665 is the Euler’s constant,l is thelength of the dipole,ae is the equivalent radius of the dipole, Si(x) is thesine integral and Ci(x) is the cosine integral given by

Si(x) =∫ x

0

sin(t)t

dt, (6.3a)

Ci(x) =∫ x

∞

cos(t)t

dt. (6.3b)

86


The real and imaginary parts of the input impedance referredto at the cur-rent at the input terminals are computed from

Rin =Rm

sin2(kl/2), (6.4a)

Xin =Xm

sin2(kl/2). (6.4b)

6.1.2 TLM and TLMME computations

The setup for the TLM simulation with absorbing boundary condition (ABC)and the TLMME simulation is shown in Figure 6.2. The dipole isoriented

Absorbing boundary

Multipole expansion boundary

radius r

Figure 6.2: Flat dipole antenna simulation setup for TLM simulation withabsorbing boundary condition and TLMME boundary condi-tion.

along thez-axis and may be considered as rotationally symmetric. Conse-quently, only TMn0 radiation modes need to be considered. These modeshave fields independent ofϕ.

87


The dipole antenna was discretized with a uniform spatial discretizetionstep∆x= ∆y= ∆z= 1mm. For the TLM simulation with ABC the dimen-sions of the computational box are 70mm×70mm×70mm resulting in atotal number of 343000 TLM cells. For the TLMME simulation the radiusr = 35mm and the number of TMn0 modesN = 5. Both, the TLM withABC and the TLMME simulations, were performed withn = 5000 timesteps.

The antenna was excited with a discrete Dirac impulse on a discretesource port distributed over 2 cell interfaces. The reference resistance ofthe source port equals377

2 Ω. The 5×5 scattering matrix of the connectionnetwork reads

S=13

−1 1 1 1 12 −2 1 1 12 1 −2 1 12 1 1 −2 12 1 1 1 −2

. (6.5)

It can be verified by direct computation that this scatteringmatrix is en-ergy conservative, i.e.,|det(S)| = 1, and thatS·S= I . The matrix is notsymmetric since the reference resistances on the ports differ.

The results of the calculated input admittance by the TLM andthe TLMMEare shown in Figure 6.3 and Figure 6.4. The results are compared with a re-sult obtained by method of moments (MoM). The MoM computation wasdone using EMAP5 software [42].

We can see a good agreement between the different methods. The ad-mittance computed by the TLMME method agrees slightly better with theMoM computation than the TLM computation with ABC.

Furthermore, in contrast to the TLM calculation with ABC, the TLMMEcomputation provides us with novel information about the radiated field,e.g., the amount of radiated energy and the distribution of this energyamong the different radiation modes. This kind of information is neveravailable when using TLM with ABC. Also, since the temporal depen-dence of the modal coefficients is known in the TLMME method, we canreconstruct the radiated field without the need to re-simulate the wholeproblem.

88


0 1e+09 2e+09 3e+09 4e+09 5e+09Frequency [Hz]

-0.005

0

0.005

0.01

0.015

0.02

Rea

l par

t of i

nput

adm

ittan

ce [1

/Ohm

]

TLMMETLM with ABCsMoM

Figure 6.3: Real part of the input admittance of the flat dipole antenna.

0 1e+09 2e+09 3e+09 4e+09 5e+09Frequency [Hz]

-0.005

0

0.005

0.01

Imag

inar

y pa

rt o

f inp

ut a

dmitt

ance

[1/O

hm] TLMME

TLM with ABCsMoM

Figure 6.4: Imaginary part of the input admittance of the flatdipole an-tenna.

89


In Figure 6.5 we can see the instantaneous energy of the incoming, re-flected and transmitted (radiated) signals. A detailed viewon the energyof reflected signal is shown in Figure 6.6. We see that the amount of en-ergy reflected from the radiation modes back to the TLM domainis muchsmaller compared to the amount of energy transmitted to the terminationimpedances of the radiation modes. Consequently, most of the energy isradiated. This explains why the results obtained by TLM withABC andby TLMME are similar in this particular example.

50 75 100 125 150

Time step [-]

0

0.002

0.004

0.006

0.008

0.01

Sig

nal e

nerg

y

Incoming energy to radiation modesReflected energy from radiation modesRadiated energy in radiation modes

Figure 6.5: Energy of incoming, reflected and radiated signals.

6.2 Dipole Antenna at the Boundary ofSimulation Region

For the absorbing boundary condition to work properly, there must be a cer-tain distance between the geometric object and the computational bound-ary. In this example a dipole antenna of 6mm length is placed at the bound-ary of the simulation region and its input impedance is computed.

90

6.2 Dipole Antenna at the Boundary of Simulation Region

50 75 100 125 150

Time step [-]

0

0.0001

0.0002

0.0003

0.0004

0.0005S

igna

l ene

rgy

Incoming energy to radiation modesReflected energy from radiation modesRadiated energy in radiation modes

Figure 6.6: Detailed view on the energy of the reflected signal.

For the comparison the following simulation setups are used:

1. Reference simulation with ABC,

2. TLM with ABC on cubical simulation region,

3. TLM with ABC on spherical TLM region,

4. TLMME boundary condition on spherical TLM region.

The discretized dipole antenna inside the spherical TLM region is shownin Figure 6.7. Except in the case of the reference simulation, the dipoleantenna is touching directly the boundary of the simulationregion.

For the reference simulation a simulation domain of the sizeof 50mm×50mm× 50mm was used. The other parameters were the same as thosedescribed below for the other simulations.

The setup for the ABC and TLMME simulations was:

• Simulation domain of 20mm×20mm×20mm,

91


Figure 6.7: Dipole antenna inside spherical TLM region. Theantenna istouching the boundary of the simulation region.

• Uniform spatial discretization with∆x= ∆y= ∆z= 1mm,

• Number of TLM iterationsn= 500 time steps,

• Radius of the spherical TLM regionr = 10mm,

• Excitation by a discrete Dirac impulse on a discrete source port dis-tributed over 2 cells with reference port resistance equal 377/2Ω(same as in the previous example).

For the TLMME simulation only TMn0 modes were considered and thenumber of modes wasN = 25.

The results of the computed input impedance for the different setupsare shown in Figure 6.8. We can see that the results of ABC simulationswhen the dipole is touching the boundary are not correct in both cases, thecubical simulation region and the spherical simulation region. The reactive

92

6.3 Bowtie Antenna

part of the input impedance does not properly reflect the resonance of theantenna. On the other hand, we can see a good agreement between the

5e+09 1e+10 1.5e+10 2e+10 2.5e+10 3e+10frequency [Hz]

-400

-200

0

200

Re/

ImZ

in

[Ω]

TLMME in spherical TLM region

ABC in spherical TLM region

ABC in cubical TLM region

ABC - reference simulation

Figure 6.8: Real and imaginary parts of the input impedance of the 6mmdipole for different simulation setups.

TLMME simulation and the reference simulation, even if the antenna istouching the boundary of the simulation region. The imaginary part reflectscorrectly the resonances of the antenna and also the real part agrees wellwith the reference simulation.

6.3 Bowtie Antenna

In this example the input impedance of broadband planar bowtie antennais computed. The purpose of this example is to demonstrate the perfor-mance of the developed excitation techniques by means of interconnectionnetworks (see Section 2.4).

The dimensions of the antenna structure are shown in Figure 6.9. As

93


was the case in the previous section, also here for the bowtieantenna theinput impedance is the main antenna characteristics of interest.

30

3090

Figure 6.9: The bowtie antenna; all dimensions are given in mm.

The bowtie antenna was discretized with a spatial discretizetion step∆l = 0.25mm in a computational box with dimensions 50mm×50mm×50mm. At the outer boundary of the simulation domain absorbing bound-ary conditions were applied.

For the excitation a discrete power source with a connectionnetworkdistributed across 2×2 cell interfaces was used. The antenna was excitedwith a Dirac impulse. The WDF representation of the connection networkis shown in Figure 6.10. The port impedances of the ports werechosen sothat realizability is ensured (see Section 5.2.2 and [25], pp. 271–272). The9×9 scattering matrix of the connection network reads

S= −14

0 V1T V1

T

2V1 M1 M2

2V1 M2 M1

, (6.6a)

whereV1 =

[

1 1 1 1]

T , (6.6b)

M1 =12

3 −5 3 3−5 3 3 33 3 3 −53 3 −5 3

(6.6c)

94

6.3 Bowtie Antenna

377/2

377

377

377/2

377/2

377/2

377/2

377

377

377

377

Source port

Figure 6.10: WDF representation of a 2× 2 connection network; the portimpedances are also shown.

95


and

M2 = −12

1 1 1 11 1 1 11 1 1 11 1 1 1

. (6.6d)

It can be checked by direct computation thatS·S= I and|det(S)| = 1.The result of the computed input impedance using the discrete source in

TLM is shown in Figure 6.11. We can see a good agreement with methodof moments (MoM) result presented in [43] and measurement result pre-sented in [44].

50 100 150 200 250Monopole length [Degree]

-300

-200

-100

0

100

200

Res

ista

nce/

Rea

ctan

ce [O

hm]

Resistance - TLM simulationResistance - MoM simulationResistance - measured dataReactance - TLM simulationReactance - MoM simulationReactance - measured data

Figure 6.11: Input impedance of the bowtie antenna. MoM simulation re-sults are taken from [43] and measurement results from [44].

In Figure 6.12 the first 150 time steps of the reflected wave at the sourceport are shown. From the returns to zero of the signal can be clearly seenthat non-physical modes are present in the simulation. However, since the

96

6.3 Bowtie Antenna

TLM model is linear and since there is only one source port, the results forlow frequency components are correct.

0 50 100 150Time steps [-]

-0.05

0

0.05

0.1

Ref

lect

ed v

olta

ge w

ave

ampl

itude

[V]

Figure 6.12: Time domain signal for 150 time steps of the reflected waveat the source port.

97

7 Conclusions

In this thesis the Transmission Line Matrix — Multipole Expansion method(TLMME) has been presented. The TLMME method allows an efficientand potentially exact modeling of radiating electromagnetic structures.

In TLMME the time-domain Transmission Line Matrix (TLM) methodis combined with the multipole expansion of the radiated field. The totalradiated field is decomposed into orthonormal radiation modes which areconnected to the TLM simulation domain on a spherical boundary. In aglobal network model the simulation domain is modeled by theTLM meshof transmission lines, every impedance of the radiation mode is modeledby a ladder network oneport and the connection of these partial networksis accomplished by an ideal transformer connection network. This allowsto include potentially exact radiating boundary conditioninto the TLMmodel by lumped element equivalent circuits representing the impedancesof radiation modes.

In Chapter 2 the principles of the TLM method were introduced. It wasshown how the scattering matrix of the TLM method is obtainedusing dis-crete differential forms. Furthermore, novel techniques for the excitation ofTLM cells by means of discrete sources were developed. The connectionnetworks employed provide as port signals incoming and outgoing wavesand are thus easily coupled to the TLM method which uses wave quantitiesas well. Also, they enable the excitation of the TLM model with a discreteDirac impulse signal. Very good performance of the discreteexcitationtechnique was verified on an example of bowtie antenna in Section 6.3.

In Chapter 3 the theory of multipole expansion and radiationmodes wasgiven. It was shown how the impedances of radiation modes canbe repre-sented using equivalent Cauer circuits.

In Chapter 4 the connection subnetwork connecting the TLM region tothe impedances of the radiation modes was described. The scattering ma-

99

7 Conclusions

trix of the connection subnetwork was derived and its special cases werediscussed.

The TLMME method was described in detail in Chapter 5. The spher-ical TLM region was described and its generation by means of Cartesianmesh generator was explained. The performance of the Cartesian meshgenerator was evaluated on an example of a rectangular cavity resonator.We could calculate the resonance frequencies with a relative error below0.2 % for the highest mesh resolution. Furthermore, the realizations of theimpedances of the radiation modes by means of wave digital filters (WDFs)were described and the details of the TLMME algorithm were explained indetail.

The results of the calculation of the input impedance of antennas usingthe TLMME method were given in Chapter 6. First, a flat dipole antennawas characterized. We have seen good agreement between the results ob-tained by the TLMME method, the TLM method using absorbing bound-ary condition (ABC) and the method of moments (MoM). However, incontrast to the TLM method with ABC, the availability of additional infor-mation about the radiated field in the TLMME method could explain whythe solution using TLM with ABC is similar to that of TLMME fortheparticular example. For the example of the flat dipole antenna most of thefield hitting the boundary of the simulation domain is radiated, as could beobserved from the energy transmitted to the termination impedances of theradiation modes.

In the example of a dipole antenna located at the boundary of the sim-ulation domain was shown that the TLMME method gives better result ofthe input impedance of the antenna than the TLM method with ABC. Incontrast to the TLM method with ABC the input impedance computed bythe TLMME method was correct even if the dipole antenna is touching theboundary of the simulation domain.

There still remain many tasks which could be done in the future. Oneproblem of the TLMME method in the current implementation isthe non-conformity of the discretized spherical manifold to the true spherical man-ifold. When a tetrahedral implementation of TLM is available, the per-formance of the TLMME method could be tested with a sphericalTLMsimulation region conforming better to the true spherical manifold.

100

Furthermore, planar structures with conducting ground area set of veryimportant problems in today’s industry. To solve the radiation problems ofthese planar structures efficiently by TLMME, a half-space formulation ofthe TLMME could be developed and implemented.

101

A Useful Formulas andRelations

A.1 Central Difference Scheme

Let’s have a functionf (x) of x ∈ R and f : R → R. The first centraldifference, using discretization step∆x, is given by

[

df (x)dx

]

D=∆ f (x)∆x

=f (x+ ∆x

2 )− f (x− ∆x2 )

∆x= f ′∆(x). (A.1)

The second order central difference is then obtained by

[

d2 f (x)

dx2

]

D=

[

ddx

([

df (x)dx

]

D

)]

D

=

[

ddx

(

∆ f (x)∆x

)]

D=

[

ddx

f ′∆(x)

]

D=

=f ′∆(x+ ∆x

2 )− f ′∆(x− ∆x

2 )

∆x=

=1

∆x2

(

f (x+∆x)−2 f (x)+ f (x−∆x))

=

= f ′′∆

(x)

(A.2)

A.2 Transformations Between Wave andNetwork Quantities

Possible transformations from network quantities to wave quantitiesTW

and from wave quantities to network quantitiesTN.

103

A Useful Formulas and Relations

Transformation 1

TW =1√

2

[

1 η

1 −η

]

(A.3a)

TN =1√

2

[

1 11η−1η

]

(A.3b)

Transformation 2

TW =

[

1 η

1 −η

]

(A.4a)

TN =12

[

1 11η−1η

]

(A.4b)

Transformation 3

TW =12

[

1 η

1 −η

]

(A.5a)

TN =

[

1 11η−1η

]

(A.5b)

Normalized wave quantities

V′ =V√η, I ′ = I

√η (A.6)

z=V′

I ′=

1η

VI=

Zη

(A.7)

Then, transformation 1 reads

TW =1√

2

[

1 11 −1

]

(A.8a)

TN =1√

2

[

1 11 −1

]

= TW (A.8b)

104

A.3 Matrix Algebra Rules

A.3 Matrix Algebra Rules

A.3.1 Transposition and inversion

(AT)T = A (A.9)

(A ±B)T = AT ±BT (A.10)

(AB)T = BTAT (A.11)

(ABC)T = (BC)TAT = CTBTAT (A.12)

For any nonsingular matrixA

(AT)−1 = (A−1)T (A.13)

A.3.2 Block matrix and Schur complement

In the following consider the matrices [A]p×p, [B]p×q, [C]q×p, [D]q×q,whereD is nonsingular. Let [M ](p+q)×(p+q) be the block matrix (partitionedmatrix)

M =[

A BC D

]

. (A.14)

The transpose ofM is given by

MT =

[

A BC D

]T

=

[

AT CT

BT DT

]

. (A.15)

The Schur complementSA is defined as

SA = A −BD−1C. (A.16)

With the Schur complement the inverse matrixM−1 can be calculated as

[

A BC D

]−1

=

[

I 0−D−1C I

] [

S−1A 00 D−1

][

I −BD−1

0 I

]

. (A.17)

105

A Useful Formulas and Relations

A.4 Taylor Expansion in 2D

The Taylor expansion of a two-dimensional functionf (x+∆x,y+∆y) aroundthe point (x,y) is given by

f (x+∆x,y+∆y) = f (x,y)+(

(∆x) fx+ (∆y) fy)

+

+12

(

(∆x)2 fxx+2∆x∆y fxy+ (∆y)2 fyy

)

+ . . . ,(A.18)

with fx = ∂ f (x,y)/∂x, fxx = ∂2 f (x,y)/∂x2, etc.

106

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111

Documents

Lehrstuhl fu¨r Hochfrequenztechnik Petr LorenzThe TLM method, as considered here, is a numerical time-domain tech- nique which has been used since its introduction by Johns and Beurle